{"id":625698,"date":"2023-09-08T01:41:32","date_gmt":"2023-09-08T01:41:32","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=625698"},"modified":"2023-09-08T01:44:19","modified_gmt":"2023-09-08T01:44:19","slug":"kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","title":{"rendered":"KC Sinha: Exercise 8.4 – Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f"},"content":{"rendered":"\n\n\n\n\n

Type-I : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u090f\u0935\u0902 \u092c\u0940\u091c\u0940\u092f \u0938\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u0940\u0927\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 :<\/strong><\/p>\n\n\n\n

Question 1<\/h4>\n\n\n\n

\u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0930\u0947\u0902 :<\/strong><\/p>\n\n\n\n

(i) $\\sin ^{2} \\theta+\\cos ^{2} \\theta=\\ldots \\ldots \\ldots .$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin ^{2} \\theta+\\cos ^{2} \\theta=1$<\/p>\n\n\n\n

(ii) $1+\\tan ^{2} \\theta=$….<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$1+\\tan ^{2} \\theta=\\sec ^{2} \\theta$<\/p>\n\n\n\n

(iii) $\\sin \\theta \\cdot \\cot \\theta$ \u0915\u093e \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e =……..<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin \\theta \\times \\frac{\\cos \\theta}{\\sin \\theta}=\\sec \\theta$<\/p>\n\n\n\n

(iv) $1-\\ldots \\ldots . . .=\\cos ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$1-\\sin ^{2} \\theta=\\cos ^{2} \\theta$<\/p>\n\n\n\n

(v) $\\tan \\mathrm{A}=\\frac{\\ldots \\ldots \\ldots}{\\cos \\mathrm{A}}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\tan A=\\frac{\\sin A}{\\cos A}$<\/p>\n\n\n\n

(vi) ….$=\\frac{\\cos A}{\\sin A}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\cot A=\\frac{\\cos }{\\sin A}$<\/p>\n\n\n\n

(vii) cos \u03b8 \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0939\u0948…….\u0915\u093e \u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\cos \\theta=\\sec \\theta$<\/p>\n\n\n\n

(viii) sin \u03b8 \u0915\u093e \u0935\u094d\u092f\u0941\u0924\u094d\u0915\u094d\u0930\u092e \u0939\u0948…….\u0915\u093e \u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin \\theta=\\operatorname{cosec} \\theta$<\/p>\n\n\n\n

(ix) sin \u03b8 \u0915\u093e \u092e\u093e\u0928 cos \u03b8=q, \u0924\u094b p \u0914\u0930 q \u092e\u0947\u0902 \u0915\u094d\u092f\u093e \u0938\u092e\u094d\u092c\u0928\u094d\u0927 \u0939\u0948?<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin \\theta=\\sqrt{1-\\cos ^{2} \\theta}$<\/p>\n\n\n\n

(x) cos \u03b8 \u0915\u093e \u092e\u093e\u0928 sin \u03b8 \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 …. \u0939\u0948\u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\cos \\theta=\\sqrt{1-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

Question 2<\/h4>\n\n\n\n

\u092f\u0926\u093f sin \u03b8=p \u0914\u0930 cos \u03b8=q, \u0924\u094b p \u0914\u0930 q \u092e\u0947\u0902 \u0915\u094d\u092f\u093e \u0938\u092e\u094d\u092c\u0928\u094d\u0927 \u0939\u0948\u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin ^{2} \\theta+\\cos ^{2} \\theta=p^{2}+q^{2}$<\/p>\n\n\n\n

$1=p^{2}+q^{2}$<\/p>\n\n\n\n

$p^{2}+q^{2}=1$<\/p>\n\n\n\n

Question 3<\/h4>\n\n\n\n

\u092f\u0926\u093f cos A=x, \u0924\u094b sin A \u0915\u094b x \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930\u0947\u0902\u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin A=\\sqrt{1-\\cos ^{2} A}=\\sqrt{1-x^{2}}$<\/p>\n\n\n\n

Question 4<\/h4>\n\n\n\n

\u092f\u0926\u093f $x \\cos \\theta=1$ \u0914\u0930 $y \\sin \\theta=1$, \u0924\u094b $\\tan \\theta$ \u0915\u093e \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0947\u0902 \u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\mathrm{Y} \\sin \\theta=1, x \\cos \\theta=1$<\/p>\n\n\n\n

\u092d\u093e\u0917 \u0926\u0947\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\frac{y \\sin \\theta}{x \\cos \\theta}=\\frac{1}{1}$<\/p>\n\n\n\n

$\\frac{x}{y} \\tan \\theta=1$<\/p>\n\n\n\n

$\\tan \\theta=\\frac{x}{y}$<\/p>\n\n\n\n

Question 5<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\cos 40^{\\circ}=p$ \u0924\u092c $\\sin 40^{\\circ}$ \u0915\u093e \u092e\u093e\u0928 p \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 \u0932\u093f\u0916\u0947\u0902 \u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin 40^{\\circ}=\\sqrt{1-\\cos ^{2} 40^{\\circ}}$<\/p>\n\n\n\n

$=\\sqrt{1-p^{2}}$<\/p>\n\n\n\n

Question 6<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\sin 77^{\\circ}=x$, \u0924\u094b $\\cos 77^{\\circ}$ \u0915\u093e \u092e\u093e\u0928 x \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 \u0932\u093f\u0916\u0947\u0902 \u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\cos 77^{\\circ}=\\sqrt{1-\\sin ^{2} 77^{\\circ}}$<\/p>\n\n\n\n

$=\\sqrt{1-x^{2}}$<\/p>\n\n\n\n

Question 7<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\cos 55^{\\circ}=x^{2}$, \u0924\u092c $\\sin 55^{\\circ}$ \u0915\u093e \u092e\u093e\u0928 x \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 \u0932\u093f\u0916\u0947\u0902<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin 55^{\\circ}=\\sqrt{1-\\cos ^{2} 55^{\\circ}}$<\/p>\n\n\n\n

$=\\sqrt{1-\\left(x^{2}\\right)^{2}}$<\/p>\n\n\n\n

$=\\sqrt{1-x^{4}}$<\/p>\n\n\n\n

Question 8<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\sin 50^{\\circ}=a$. \u0924\u092c $\\cos 50^{\\circ}$ \u0915\u093e \u092e\u093e\u0928 a \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 \u0932\u093f\u0916\u0947\u0902 \u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\cos 50^{\\circ}=\\sqrt{1-\\sin ^{2} 50^{\\circ}}$<\/p>\n\n\n\n

$=\\sqrt{1-a^{2}}$<\/p>\n\n\n\n

Question 9<\/h4>\n\n\n\n

\u092f\u0926\u093f xcos A=1 \u0914\u0930 tan A=y, \u0924\u092c $x^{2}-y^{2}$ \u0915\u093e \u092e\u093e\u0928 \u0915\u094d\u092f\u093e \u0939\u0948 \u0964<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$X=\\frac{1}{\\cos A}=x=\\sec A$<\/p>\n\n\n\n

$Y=\\tan A$<\/p>\n\n\n\n

$x^{2}-y^{2}=\\sec ^{2} \\mathrm{~A}-\\tan ^{2} \\mathrm{~A}$<\/p>\n\n\n\n

=1<\/p>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0964<\/strong><\/p>\n\n\n\n

Question 10<\/h4>\n\n\n\n

(i) $(1-\\sin \\theta)(1+\\sin \\theta)=\\cos ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$(1-\\sin \\theta) \\times(1+\\sin \\theta)$<\/p>\n\n\n\n

$1^{2}-\\sin ^{2} \\theta$<\/p>\n\n\n\n

$1-\\sin ^{2} \\theta$<\/p>\n\n\n\n

$=\\cos ^{2} \\theta$<\/p>\n\n\n\n

(ii) $(1+\\cos \\theta)(1-\\cos \\theta)=\\sin ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$(1+\\cos \\theta) \\times(1-\\cos \\theta)$<\/p>\n\n\n\n

$1^{2}-\\cos ^{2} \\theta$<\/p>\n\n\n\n

$1-\\cos ^{2} \\theta$<\/p>\n\n\n\n

$=\\sin ^{2} \\theta$<\/p>\n\n\n\n

Question 11<\/h4>\n\n\n\n

$\\frac{(1-\\cos \\theta)(1+\\cos \\theta)}{(1-\\sin \\theta)(1+\\sin \\theta)}=\\tan ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{(1-\\cos \\theta)(1+\\cos \\theta)}{(1-\\sin \\theta)(1+\\sin \\theta)}$<\/p>\n\n\n\n

$\\frac{1^{2}-\\cos ^{2} \\theta}{1^{2}-\\sin ^{2} \\theta}=\\frac{1-\\cos ^{2} \\theta}{1-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\theta}{\\cos ^{2} \\theta}=\\tan ^{2} \\theta$<\/p>\n\n\n\n

Question 12<\/h4>\n\n\n\n

$\\frac{1}{\\sec \\theta+\\tan \\theta}=\\sec \\theta-\\tan \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$=\\frac{\\sec ^{2} \\theta-\\tan ^{2} \\theta}{\\sec \\theta+\\tan \\theta}=\\frac{(\\sec \\theta-\\tan \\theta)(\\sec \\theta+\\tan \\theta)}{\\sec \\theta+\\tan \\theta}$<\/p>\n\n\n\n

$=\\sec \\theta-\\tan \\theta=$ R.H.S<\/p>\n\n\n\n

Question 13<\/h4>\n\n\n\n

$\\frac{\\sin ^{3} \\theta+\\cos ^{3} \\theta}{\\sin \\theta+\\cos \\theta}+\\sin \\theta \\cos \\theta=1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\frac{\\sin ^{3} \\theta+\\cos ^{3} \\theta}{\\sin \\theta+\\cos \\theta}+\\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

$a^{3}+b^{3}=(\\mathrm{a}+\\mathrm{b})\\left(a^{2}-a b+b^{2}\\right)$<\/p>\n\n\n\n

$\\frac{(\\sin \\theta+\\cos \\theta)\\left(\\sin ^{2} \\theta-\\sin \\theta \\times \\cos \\theta+\\cos ^{2}\\right)}{\\sin \\theta+\\cos \\theta}+\\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

$\\sin ^{2} \\theta-\\sin \\theta \\times \\cos \\theta+\\cos ^{2} \\theta+\\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

$\\sin ^{2} \\theta+\\cos ^{2} \\theta-\\sin \\theta \\times \\cos \\theta+\\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

$1-\\sin \\theta \\times \\cos \\theta+\\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

=1 RHS<\/p>\n\n\n\n

Type II : LCM \u0932\u0947\u0915\u0930 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 :<\/strong><\/p>\n\n\n\n

Question 14<\/h4>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 :<\/strong><\/p>\n\n\n\n

(i) $\\sin \\theta \\cdot \\cot \\theta=\\cos \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sin \\theta \\times \\cot \\theta$<\/p>\n\n\n\n

$\\sin \\theta \\times \\frac{\\cos \\theta}{\\sin \\theta}=\\cos \\theta$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

(ii) $\\sin ^{2} \\theta\\left(1+\\cot ^{2} \\theta\\right)=1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sin ^{2} \\theta\\left(1+\\cot ^{2} \\theta\\right)$<\/p>\n\n\n\n

$\\sin ^{2} \\theta \\times \\operatorname{cosec}^{2} \\theta=(\\sin \\theta \\times \\operatorname{cosec} \\theta)^{2}$<\/p>\n\n\n\n

$(1)^{2}=1$=RHS<\/p>\n\n\n\n

(iii) $\\cos ^{2} \\mathrm{~A}\\left(\\tan ^{2} \\mathrm{~A}+1\\right)=1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\cos ^{2} A\\left(\\tan ^{2} A+1\\right)$<\/p>\n\n\n\n

$\\cos ^{2} A \\times \\sec ^{2} A$<\/p>\n\n\n\n

$(\\cos A \\times \\sec A)^{2}$<\/p>\n\n\n\n

$(1)^{2}=1$ RHS<\/p>\n\n\n\n

(iv) $\\tan ^{4} \\theta+\\tan ^{2} \\theta=\\sec ^{4} \\theta-\\sec ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\tan ^{4} \\theta+\\tan ^{2} \\theta$<\/p>\n\n\n\n

$\\left(\\tan ^{2} \\theta\\right)^{2}+\\tan ^{2} \\theta$<\/p>\n\n\n\n

$\\left(\\sec ^{2} \\theta-1\\right)^{2}+\\sec ^{2} \\theta-1$<\/p>\n\n\n\n

$\\left(\\sec ^{2} \\theta\\right)^{2}-2 \\sec ^{2} \\theta \\times 1+(1)^{2}+\\sec ^{2} \\theta-1$<\/p>\n\n\n\n

$\\sec ^{4} \\theta- 2\\sec ^{2} \\theta+1+\\sec ^{2} \\theta-1$\u263a<\/p>\n\n\n\n

$\\sec ^{4} \\theta-\\sec ^{4} \\theta$<\/p>\n\n\n\n

(v) $\\frac{\\left(1+\\tan ^{2} \\theta\\right) \\sin ^{2} \\theta}{\\tan \\theta}=\\tan \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{\\left(1+\\tan ^{2} \\theta\\right) \\sin ^{2}}{\\tan \\theta}$<\/p>\n\n\n\n

$\\frac{\\sec ^{2} \\theta \\times \\sin ^{2} \\theta}{\\tan \\theta}$<\/p>\n\n\n\n

$\\frac{\\frac{1}{\\cos ^{2} \\theta} \\times \\sin ^{2} \\theta}{\\frac{\\sin \\theta}{\\cos \\theta}}$<\/p>\n\n\n\n

$=\\frac{1}{\\cos \\theta} \\times \\sin \\theta=\\frac{\\sin \\theta}{\\cos \\theta}=\\tan \\theta$<\/p>\n\n\n\n

(vi) $\\frac{\\sin ^{2} \\theta}{\\cos ^{2} \\theta}+1=\\frac{\\tan ^{2} \\theta}{\\sin ^{2} \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\theta}{\\cos ^{2} \\theta}+1$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\theta+\\cos ^{2} \\theta}{\\cos ^{2} \\theta}=\\frac{1}{\\cos ^{2} \\theta}=\\operatorname{sec}^{2} \\theta$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

$\\frac{\\tan ^{2} \\theta}{\\sin ^{2} \\theta}=\\frac{\\sin ^{2} \\theta}{\\frac{\\cos ^{2} \\theta}{\\sin ^{2} \\theta}}$<\/p>\n\n\n\n

$=\\frac{1}{\\cos ^{2} \\theta}=\\sec ^{2} \\theta$<\/p>\n\n\n\n

LHS=RHS<\/p>\n\n\n\n

(vii) $\\frac{3-4 \\sin ^{2} \\theta}{\\cos ^{2} \\theta}=3-\\tan ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{3-4 \\sin ^{2} \\theta}{\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{3}{\\cos ^{2} \\theta}-\\frac{4 \\sin ^{2} \\theta}{\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$3 \\sec ^{2} \\theta-4 \\tan ^{2} \\theta$<\/p>\n\n\n\n

$3 \\tan ^{2} \\theta-4 \\tan ^{2} \\theta$<\/p>\n\n\n\n

$3-\\tan ^{2} \\theta $ R.H.S<\/p>\n\n\n\n

(viii) $\\left(1+\\tan ^{2} \\theta\\right) \\cos \\theta \\cdot \\sin \\theta=\\tan \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\left(1+\\tan ^{2} \\theta\\right) \\cos \\theta \\cdot \\sin \\theta$<\/p>\n\n\n\n

$\\sec ^{2} \\theta \\times \\cos \\theta \\times \\sin \\theta$<\/p>\n\n\n\n

$\\frac{1}{\\cos ^{2} \\theta} \\times \\cos \\theta \\times \\sin \\theta$<\/p>\n\n\n\n

$\\frac{\\sin \\theta}{\\cos \\theta}=\\tan \\theta$ <\/p>\n\n\n\n

R.H.S<\/p>\n\n\n\n

(ix) $\\sin ^{2} \\theta-\\cos ^{2} \\phi=\\sin ^{2} \\phi-\\cos ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sin ^{2} \\theta-\\cos ^{2} \\phi$<\/p>\n\n\n\n

$\\left(1-\\cos ^{2} \\theta\\right)-\\left(1-\\sin ^{2} \\phi\\right)$<\/p>\n\n\n\n

1- $\\cos ^{2} \\theta-1+\\sin ^{2} \\phi$<\/p>\n\n\n\n

$-\\cos ^{2} \\theta+\\sin ^{2} \\phi$<\/p>\n\n\n\n

$\\sin ^{2} \\phi-\\cos ^{2} \\theta $ RHS<\/p>\n\n\n\n

(x) $\\frac{1-\\tan ^{2} \\theta}{\\cot ^{2} \\theta-1}=\\tan ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{1-\\tan ^{2} \\theta}{\\cot ^{2} \\theta-1}$<\/p>\n\n\n\n

$\\frac{1-\\frac{\\sin ^{2} \\theta}{\\cos ^{2} \\theta}}{\\frac{\\cos ^{2}}{\\sin ^{2}}-1}$<\/p>\n\n\n\n

$\\frac{\\frac{\\cos ^{2} \\theta-\\sin ^{2} \\theta}{\\cos ^{2} \\theta}}{\\frac{\\cos ^{2} \\theta-\\sin ^{2} \\theta}{\\sin ^{2} \\theta}}$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\theta}{\\cos ^{2} \\theta}=\\tan ^{2} \\theta$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

Question 15<\/h4>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 :<\/strong><\/p>\n\n\n\n

(i) $(1-\\cos \\theta)(1+\\cos \\theta)\\left(1+\\cot ^{2} \\theta\\right)=1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$=(1-\\cos \\theta)(1+\\cos \\theta)\\left(1+\\cot ^{2} \\theta\\right)$<\/p>\n\n\n\n

$=\\left(1-\\cos ^{2} \\theta\\right)\\left(1+\\cot ^{2} \\theta\\right)$<\/p>\n\n\n\n

$=\\sin ^{2} \\theta \\times \\operatorname{cosec}^{2} \\theta$<\/p>\n\n\n\n

=1<\/p>\n\n\n\n

(ii) $\\frac{(1+\\sin \\theta)^{2}+(1-\\sin \\theta)^{2}}{2 \\cos ^{2} \\theta}=\\frac{1+\\sin \\theta}{1-\\sin ^{2} \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{(1+\\sin \\theta)^{2}+(1-\\sin \\theta)^{2}}{2 \\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{\\left(1^{2}+\\sin ^{2} \\theta+2 \\times 1 \\times \\sin \\theta\\right)+\\left(1^{2}+\\sin ^{2} \\theta-2 \\times 1 \\times \\sin \\theta\\right)}{2 \\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{1+\\sin ^{2} \\theta+2 \\sin \\theta+1+\\sin ^{2} \\theta-2 \\sin \\theta}{2 \\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{2+2 \\sin ^{2} \\theta}{2 \\cos ^{2} \\theta}=\\frac{2\\left(1+\\sin ^{2} \\theta\\right)}{2\\left(1-\\sin ^{2} \\theta\\right)}$<\/p>\n\n\n\n

$\\frac{1-\\sin ^{2} \\theta}{1+\\sin ^{2} \\theta}$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

(iii) $\\frac{\\cos ^{2} \\theta(1-\\cos \\theta)}{\\sin ^{2} \\theta(1-\\sin \\theta)}=\\frac{1+\\sin \\theta}{1+\\cos \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{\\cos ^{2} \\theta(1-\\cos \\theta)}{\\sin ^{2} \\theta(1-\\sin \\theta)}$<\/p>\n\n\n\n

$\\frac{\\left(1-\\sin ^{2} \\theta\\right)(1-\\cos \\theta)}{\\left(1-\\cos ^{2} \\theta\\right)(1-\\sin \\theta)}$<\/p>\n\n\n\n

$\\frac{(1-\\sin \\theta)(1+\\sin )(1-\\cos \\theta)}{(1-\\operatorname{cos} \\theta)(1+\\cos \\theta)(1-\\sin \\theta)}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta}{1+\\cos \\theta}$ RHS<\/p>\n\n\n\n

(iv) $(\\sin \\theta-\\cos \\theta)^{2}=1-2 \\sin \\theta \\cdot \\cos \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$(\\sin \\theta-\\cos \\theta)^{2}$<\/p>\n\n\n\n

$\\sin ^{2} \\theta+\\cos ^{2} \\theta-2 \\times \\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

$\\sin ^{2} \\theta+\\cos ^{2} \\theta-2 \\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

$1-2 \\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

R.H.S <\/p>\n\n\n\n

(v) $(\\sin \\theta+\\cos \\theta)^{2}+(\\sin \\theta-\\cos \\theta)^{2}=2$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$(\\sin \\theta+\\cos \\theta)^{2}+(\\sin \\theta-\\cos \\theta)^{2}=2$<\/p>\n\n\n\n

$\\sin ^{2} \\theta+\\cos ^{2} \\theta+2 \\sin \\theta \\times \\cos \\theta+\\sin ^{2} \\theta+\\cos ^{2} \\theta-2 \\sin \\theta \\times \\cos \\theta$<\/p>\n\n\n\n

$\\sin ^{2} \\theta+\\cos ^{2} \\theta+\\sin ^{2} \\theta+\\cos ^{2} \\theta$<\/p>\n\n\n\n

$2 \\sin ^{2} \\theta+2 \\cos ^{2} \\theta$<\/p>\n\n\n\n

$2\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta\\right)$<\/p>\n\n\n\n

2\u00d71=2 RHS<\/p>\n\n\n\n

(vi) $(a \\sin \\theta+b \\cos \\theta)^{2}+(a \\cos \\theta-b \\sin \\theta)^{2}=a^{2}+b^{2}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$(a \\sin \\theta+b \\cos \\theta)^{2}+(a \\cos \\theta-b \\sin \\theta)^{2}$<\/p>\n\n\n\n

$(a \\sin \\theta)^{2}+(b \\cos \\theta)^{2}+2 \\times a \\sin \\theta \\times b \\cos \\theta+(a \\cos \\theta)^{2}+(b \\sin \\theta)^{2}-2 a \\cos \\theta \\times b \\sin \\theta$<\/p>\n\n\n\n

$a^{2} \\times \\sin ^{2} \\theta+b^{2} \\times \\cos ^{2} \\theta+2 a \\sin \\theta \\times b \\cos \\theta+a^{2} \\times \\cos ^{2} \\theta+b^{2} \\times \\sin ^{2} \\theta-2 a \\cos \\theta \\times b \\sin \\theta$<\/p>\n\n\n\n

$a^{2} \\sin ^{2} \\theta+b^{2} \\cos ^{2} \\theta+a^{2} \\cos ^{2} \\theta+b^{2} \\sin ^{2} \\theta$<\/p>\n\n\n\n

$\\left(a^{2} \\sin ^{2} \\theta+a^{2} \\cos ^{2} \\theta\\right)+\\left(b^{2} \\cos ^{2} \\theta+b^{2} \\sin ^{2} \\theta\\right)$<\/p>\n\n\n\n

$a^{2}\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta\\right)+b^{2}\\left(\\cos ^{2} \\theta+\\sin ^{2} \\theta\\right)$<\/p>\n\n\n\n

$a^{2} \\times 1+b^{2} \\times 1$<\/p>\n\n\n\n

$a^{2}+b^{2}$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

(vii) $\\cos ^{4} A+\\sin ^{4} A+2 \\sin ^{2} A \\cdot \\cos ^{2} A=1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\cos ^{4} A+\\sin ^{4} A+2 \\sin ^{2} A \\times \\cos ^{2} A$<\/p>\n\n\n\n

$\\left(\\cos ^{2} A\\right)^{2}-\\left(\\sin ^{2} A\\right)^{2}+2 \\sin ^{2} A \\times \\cos ^{2} A$<\/p>\n\n\n\n

$\\left(\\cos ^{2} A+\\sin ^{2} A\\right) 2 \\sin ^{2} A \\times \\cos ^{2} A$<\/p>\n\n\n\n

$=(1)^{2}$=1 RHS<\/p>\n\n\n\n

(viii) $\\sin ^{4} A-\\cos ^{4} A=2 \\sin ^{2} A-1=1-2 \\cos ^{2} A=\\sin ^{2} A-\\cos ^{2} A$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sin ^{4} A-\\cos ^{4} A$<\/p>\n\n\n\n

$\\left(\\sin ^{2} A\\right)^{2}-\\left(\\cos ^{2} A\\right)^{2}$<\/p>\n\n\n\n

$\\left(\\sin ^{2} A+\\cos ^{2} A\\right)\\left(\\sin ^{2} A-\\cos ^{2} A\\right)$<\/p>\n\n\n\n

$=\\sin ^{2} A-\\cos ^{2} A$<\/p>\n\n\n\n

$=\\sin ^{2} A-\\left(1-\\sin ^{2} A\\right)$<\/p>\n\n\n\n

$=\\sin ^{2} A-1+\\sin ^{2} A$<\/p>\n\n\n\n

$=2 \\sin ^{2} A-1$<\/p>\n\n\n\n

$=2\\left(-\\cos ^{2} A\\right)-1$<\/p>\n\n\n\n

$=2-2 \\cos ^{2} A-1=1-2 \\cos ^{2} A$<\/p>\n\n\n\n

=RHS<\/p>\n\n\n\n

(ix) $\\cos ^{4} \\theta-\\sin ^{4} \\theta=\\cos ^{2} \\theta-\\sin ^{2} \\theta=2 \\cos ^{2} \\theta-1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\cos ^{4} \\theta-\\sin ^{4} \\theta=\\cos ^{2} \\theta-\\sin ^{2} \\theta=2 \\cos ^{2} \\theta-1$<\/p>\n\n\n\n

$\\cos ^{4} \\theta-\\sin ^{4} \\theta=\\left(\\cos ^{2} \\theta\\right)^{2}-\\left(\\sin ^{2} \\theta\\right)^{2}$<\/p>\n\n\n\n

$\\left(\\cos ^{2} \\theta+\\sin ^{2} \\theta\\right)\\left(\\cos ^{2} \\theta-\\sin ^{2} \\theta\\right)$<\/p>\n\n\n\n

$\\cos ^{2} \\theta-\\sin ^{2} \\theta$<\/p>\n\n\n\n

$\\cos ^{2} \\theta-\\left(1-\\cos ^{2} \\theta\\right)$<\/p>\n\n\n\n

$\\cos ^{2} \\theta-1+\\cos ^{2} \\theta$<\/p>\n\n\n\n

$2 \\cos ^{2} \\theta-1$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

(x) $2 \\cos ^{2} \\theta-\\cos ^{4} \\theta+\\sin ^{4} \\theta=1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$2 \\cos ^{2} \\theta-\\cos ^{4} \\theta+\\sin ^{4} \\theta$<\/p>\n\n\n\n

$2 \\cos ^{2} \\theta-\\left(\\cos ^{2} \\theta\\right)^{2}+\\left(\\sin ^{2} \\theta\\right)^{2}$<\/p>\n\n\n\n

$2 \\cos ^{2} \\theta-\\left(\\cos ^{2} \\theta+\\sin ^{2} \\theta\\right)$<\/p>\n\n\n\n

$2 \\cos ^{2} \\theta-\\cos ^{2} \\theta+\\sin ^{2} \\theta$<\/p>\n\n\n\n

$2 \\cos ^{2} \\theta-\\cos ^{2} \\theta+1-\\cos ^{2} \\theta$<\/p>\n\n\n\n

$2 \\cos ^{2} \\theta-2 \\cos ^{2} \\theta+1$<\/p>\n\n\n\n

0+1=1<\/p>\n\n\n\n

(xi) $1-2 \\cos ^{2} \\theta+\\cos ^{4} \\theta=\\sin ^{4} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$1-2 \\cos ^{2} \\theta+\\cos ^{4} \\theta$<\/p>\n\n\n\n

$1^{2}-2 \\times 1 \\times \\cos ^{2} \\theta+\\left(\\cos ^{2}\\right)^{2}$<\/p>\n\n\n\n

$\\left(1-\\cos ^{2} \\theta\\right)^{2}$<\/p>\n\n\n\n

$\\left(\\sin ^{2} \\theta\\right)^{4}=\\sin ^{4} \\theta$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

(xii) $1-2 \\sin ^{2} \\theta+\\sin ^{4} \\theta=\\cos ^{4} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$1-2 \\sin ^{2} \\theta+\\sin ^{4} \\theta$<\/p>\n\n\n\n

$1^{2}-2 \\times 1 \\times \\sin ^{2} \\theta+\\left(\\sin ^{2}\\right)^{2}$<\/p>\n\n\n\n

$\\left(1-\\sin ^{2} \\theta\\right)^{2}$<\/p>\n\n\n\n

$\\left(\\cos ^{2} \\theta\\right)^{2}=\\cos ^{4} \\theta$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

Question 16<\/h4>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 :<\/strong><\/p>\n\n\n\n

(i) $\\sec ^{2} \\theta+\\operatorname{cosec}^{2} \\theta=\\sec ^{2} \\theta \\cdot \\operatorname{cosec}^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sec ^{2} \\theta+\\operatorname{cosec}^{2} \\theta$<\/p>\n\n\n\n

$\\frac{1}{\\cos ^{2} \\theta}+\\frac{1}{\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\theta+\\cos ^{2} \\theta}{\\cos ^{2} \\theta \\times \\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{1}{\\cos ^{2} \\theta \\times \\sin ^{2} \\theta}$<\/p>\n\n\n\n

$=\\frac{1}{\\frac{1}{\\sec ^{2} \\theta} \\times \\frac{1}{\\operatorname{cosec}^{2} \\theta}}$<\/p>\n\n\n\n

$\\sec ^{2} \\theta \\times \\operatorname{cosec}^{2} \\theta$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

(ii) $\\frac{\\cos ^{2} \\theta}{\\sin \\theta}+\\sin \\theta=\\operatorname{cosec} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{\\cos ^{2} \\theta}{\\sin \\theta}+\\sin \\theta$<\/p>\n\n\n\n

$\\frac{1-\\sin ^{2} \\theta}{\\sin \\theta}+\\sin \\theta$<\/p>\n\n\n\n

$\\frac{1}{\\sin \\theta}-\\frac{\\sin ^{2} \\theta}{\\sin \\theta}+\\sin \\theta$<\/p>\n\n\n\n

$\\frac{1}{\\sin \\theta}-\\sin \\theta+\\sin \\theta$<\/p>\n\n\n\n

cosec \u03b8<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

(iii) $\\cot \\theta+\\tan \\theta=\\operatorname{cosec} \\theta \\cdot \\sec \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\cot \\theta+\\tan \\theta$<\/p>\n\n\n\n

$\\frac{\\cos \\theta}{\\sin \\theta}+\\frac{\\sin \\theta}{\\cos \\theta}$<\/p>\n\n\n\n

$\\frac{\\cos ^{2} \\theta+\\sin ^{2} \\theta}{\\sin \\theta \\times \\cos \\theta}$<\/p>\n\n\n\n

$\\frac{1}{\\sin \\theta \\times \\cos \\theta}$<\/p>\n\n\n\n

$\\frac{1}{\\frac{1}{\\sin \\theta} \\times \\frac{1}{\\cos \\theta}}$<\/p>\n\n\n\n

$\\operatorname{cosec} \\theta \\times \\sec \\theta$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

Question 17<\/h4>\n\n\n\n

$\\frac{1-\\sin \\theta}{1+\\sin \\theta}=\\left(\\frac{1-\\sin \\theta}{\\cos \\theta}\\right)^{2}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{1-\\sin \\theta}{1+\\sin \\theta}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\frac{1-\\sin \\theta}{1+\\sin \\theta} \\times \\frac{1-\\sin \\theta}{1-\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{(1-\\sin \\theta)^{2}}{(1)^{2}-(\\sin \\theta)^{2}}$<\/p>\n\n\n\n

$\\frac{(1-\\sin \\theta)^{2}}{1-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{(1-\\sin \\theta)^{2}}{\\cos ^{2} \\theta$<\/p>\n\n\n\n

$\\left(\\frac{1-\\sin \\theta}{\\cos \\theta}\\right)^{2}$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

Question 18<\/h4>\n\n\n\n

$\\frac{1-\\cos \\theta}{1+\\cos \\theta}=\\left(\\frac{1-\\cos \\theta}{\\sin \\theta}\\right)^{2}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{1-\\cos \\theta}{1+\\cos \\theta}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\frac{1-\\cos \\theta}{1+\\cos \\theta} \\times \\frac{1-\\cos \\theta}{1-\\cos \\theta}$<\/p>\n\n\n\n

$\\frac{(1-\\cos \\theta)^{2}}{(1)^{2}-(\\cos \\theta)^{2}}$<\/p>\n\n\n\n

$\\frac{(1-\\cos \\theta)^{2}}{1-\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{(1-\\cos \\theta)^{2}}{\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\left(\\frac{1-\\cos \\theta}{\\sin \\theta}\\right)^{2}$ RHS<\/p>\n\n\n\n

Question 19<\/h4>\n\n\n\n

$\\left(\\frac{1+\\cos \\theta}{\\sin \\theta}\\right)^{2}=\\frac{1+\\cos \\theta}{1-\\cos \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\left(\\frac{1+\\cos \\theta}{\\sin \\theta}\\right)^{2}$ <\/p>\n\n\n\n

$\\frac{(1+\\cos \\theta)^{2}}{\\sin ^{2} \\theta}$ <\/p>\n\n\n\n

$\\frac{(1+\\cos \\theta)^{2}}{1^{2}-\\cos ^{2} \\theta}$ <\/p>\n\n\n\n

$\\frac{(1-\\cos \\theta)^{2}}{(1+\\cos \\theta)(1-\\cos \\theta)}$ <\/p>\n\n\n\n

$\\frac{1+\\cos \\theta}{1-\\cos \\theta}$ R.H.S.<\/p>\n\n\n\n

Question 20<\/h4>\n\n\n\n

$\\frac{\\cos \\theta}{1+\\sin \\theta}=\\frac{1-\\sin \\theta}{\\cos \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\frac{\\cos \\theta}{1+\\sin \\theta}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\frac{\\cos \\theta}{1+\\sin \\theta} \\times \\frac{1-\\sin \\theta}{1-\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{\\cos \\theta(1-\\sin \\theta)^{2}}{1^{2}-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{\\cos \\theta(1-\\cos \\theta)^{2}}{\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{1-\\sin \\theta}{\\cos \\theta}$ RHS<\/p>\n\n\n\n

Question 21<\/h4>\n\n\n\n

$\\left(\\sin ^{8} \\theta-\\cos ^{8} \\theta\\right)=\\left(\\sin ^{2} \\theta-\\cos ^{2} \\theta\\right)\\left(1-2 \\sin ^{2} \\theta \\cdot \\cos ^{2} \\theta\\right)$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\left(\\sin ^{8} \\theta-\\cos ^{8} \\theta\\right)$<\/p>\n\n\n\n

$\\left[\\left(\\sin ^{2} \\theta\\right)^{2}\\right]^{2}-\\left[\\left(\\cos ^{2} \\theta\\right)^{2}\\right]^{2}$<\/p>\n\n\n\n

$\\left[\\left(\\sin ^{2} \\theta\\right)^{2}-\\left(\\cos ^{2} \\theta\\right)^{2}\\right] \\times\\left[\\left(\\sin ^{2} \\theta\\right)^{2}+\\left(\\cos ^{2} \\theta\\right)^{2}\\right]$<\/p>\n\n\n\n

$\\left[\\left(\\sin ^{2} \\theta-\\cos ^{2} \\theta\\right)\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta\\right)\\right]\\left[\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta\\right)^{2}-\\right.\\left.2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]$<\/p>\n\n\n\n

$\\left(\\sin ^{2} \\theta-\\cos ^{2} \\theta\\right) \\times 1\\left[(1)^{2}-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]$<\/p>\n\n\n\n

$\\left(\\sin ^{2} \\theta-\\cos ^{2} \\theta\\right) \\times\\left(1-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right)$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

Question 22<\/h4>\n\n\n\n

$2\\left(\\sin ^{6} \\theta+\\cos ^{6} \\theta\\right)-3\\left(\\sin ^{4} \\theta+\\cos ^{4} \\theta\\right)+1=0$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$=2\\left(\\sin ^{6} \\theta+\\cos ^{6} \\theta\\right)-3\\left(\\sin ^{4} \\theta+\\cos ^{4} \\theta\\right)+1$<\/p>\n\n\n\n

$=\\left[\\left(\\sin ^{2} \\theta\\right)^{3}+\\left(\\cos ^{2} \\theta\\right)^{3}\\right]-3\\left[\\left(\\sin ^{2} \\theta\\right)^{2}+\\left(\\cos ^{2} \\theta\\right)^{2}\\right]+1$<\/p>\n\n\n\n

$=2\\left[\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta\\right)\\left(\\sin ^{2} \\theta\\right)^{2}-\\sin ^{2} \\theta \\times \\cos ^{2} \\theta+\\left(\\cos ^{2} \\theta\\right)^{2}\\right]-3\\left[\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta\\right)^{2}-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]+1$<\/p>\n\n\n\n

$=2 \\times 1\\left[\\left(\\sin ^{2} \\theta\\right)^{2}+\\left(\\cos ^{2} \\theta\\right)^{2}-\\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]-3 \\times\\left[(1)^{2}-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]+1$<\/p>\n\n\n\n

$=2\\left[\\left(\\sin ^{2} \\theta\\right)^{2}+\\left(\\cos ^{2} \\theta\\right)^{2}-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]-3\\left[(1)^{2}-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]+1$<\/p>\n\n\n\n

$=2\\left[\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta\\right)^{2}-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta-\\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]-3\\left[1-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]+1$<\/p>\n\n\n\n

$=2\\left[(1)^{2}-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta-\\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]-3\\left[1-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]+1$<\/p>\n\n\n\n

$=2\\left[1-3 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]-3\\left[1-2 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta\\right]+1$<\/p>\n\n\n\n

$=2-6 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta-3+6 \\sin ^{2} \\theta \\times \\cos ^{2} \\theta+1$<\/p>\n\n\n\n

=2-3+1=3-3=0<\/p>\n\n\n\n

Type-III : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0935\u094d\u092f\u0902\u091c\u0915\u094b\u0902 \u0915\u0947 \u0935\u0930\u094d\u0917\u092e\u0942\u0932 \u0938\u0947 \u0938\u092e\u094d\u092c\u0926\u094d\u0927 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0928\u0947 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 :<\/strong><\/p>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 :<\/strong><\/p>\n\n\n\n

Question 23<\/h4>\n\n\n\n

$\\frac{\\cos A}{1-\\tan A}+\\frac{\\sin A}{1-\\cot A}=\\sin A+\\cos A$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{\\cos \\mathrm{A}}{1-\\tan \\mathrm{A}}+\\frac{\\sin \\mathrm{A}}{1-\\cot \\mathrm{A}}$<\/p>\n\n\n\n

$\\frac{\\cos \\mathrm{A}}{1-\\frac{\\sin \\mathrm{A}}{\\cos \\mathrm{A}}}+\\frac{\\sin \\mathrm{A}}{1-\\frac{\\cos \\mathrm{A}}{\\sin \\mathrm{A}}}$<\/p>\n\n\n\n

$\\frac{\\cos \\mathrm{A}}{\\frac{\\cos \\mathrm{A}-\\sin \\mathrm{A}}{\\cos \\mathrm{A}}}+\\frac{\\sin \\mathrm{A}}{\\frac{\\sin \\mathrm{A}-\\cos \\mathrm{A}}{\\sin \\mathrm{A}}}$<\/p>\n\n\n\n

$\\frac{\\cos ^{2} \\mathrm{~A}}{\\cos \\mathrm{A}-\\sin \\mathrm{A}}+\\frac{\\sin ^{2} \\mathrm{~A}}{\\sin \\mathrm{A}-\\cos \\mathrm{A}}$<\/p>\n\n\n\n

$\\frac{\\cos ^{2} \\mathrm{~A}}{\\cos \\mathrm{A}-\\sin \\mathrm{A}}-\\frac{\\sin ^{2} \\mathrm{~A}}{\\sin \\mathrm{A}-\\cos \\mathrm{A}}$<\/p>\n\n\n\n

$\\frac{\\cos ^{2} \\mathrm{~A}-\\sin ^{2} \\mathrm{~A}}{\\cos \\mathrm{A}-\\sin \\mathrm{A}}$<\/p>\n\n\n\n

$\\frac{(\\cos \\mathrm{A}-\\sin \\mathrm{A})(\\cos \\mathrm{A}+\\sin \\mathrm{A})}{\\cos \\mathrm{A}-\\sin \\mathrm{A}}$<\/p>\n\n\n\n

cos A+sin A <\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

Question 24<\/h4>\n\n\n\n

$\\frac{\\sin \\theta}{1+\\cos \\theta}+\\frac{1+\\cos \\theta}{\\sin \\theta}=\\frac{2}{\\sin \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{\\sin \\theta}{1+\\cos \\theta}+\\frac{1+\\cos \\theta}{\\sin \\theta}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\frac{\\sin \\theta}{1+\\cos \\theta} \\times \\frac{1-\\cos \\theta}{1-\\cos \\theta}+\\frac{1+\\cos \\theta}{\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{\\sin \\theta(1-\\cos \\theta)}{1^{2}+\\cos ^{2} \\theta}+\\frac{1+\\cos \\theta}{\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{\\sin \\theta(1-\\cos \\theta)}{\\sin ^{2} \\theta}+\\frac{1+\\cos \\theta}{\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{(1-\\cos \\theta)+1+\\cos \\theta}{\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{2}{\\sin \\theta}$ <\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

Question 25<\/h4>\n\n\n\n

$\\frac{1}{1+\\sin \\theta}+\\frac{1}{1-\\sin \\theta}=2 \\sec ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{1}{1+\\sin \\theta}+\\frac{1}{1-\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{1-\\sin \\theta+1+\\sin \\theta}{(1+\\sin \\theta)(1-\\sin \\theta)}$<\/p>\n\n\n\n

$\\frac{2}{1^{2}-\\sin ^{2} \\theta}=\\frac{2}{1-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{2}{\\cos ^{2} \\theta}=2 \\sec ^{2} \\theta$<\/p>\n\n\n\n

Proved<\/p>\n\n\n\n

Question 26<\/h4>\n\n\n\n

$\\frac{1+\\sin \\theta}{\\cos \\theta}+\\frac{\\cos \\theta}{1+\\sin \\theta}=2 \\sec \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta}{\\cos \\theta}+\\frac{\\cos \\theta}{1+\\sin \\theta}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta}{\\cos \\theta}+\\frac{\\cos \\theta}{1+\\sin \\theta} \\times \\frac{1-\\sin \\theta}{1-\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta}{\\cos \\theta}+\\frac{\\cos \\theta(1-\\sin \\theta)}{1^{2}-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta}{\\cos \\theta}+\\frac{\\cos \\theta(1-\\sin \\theta)}{1-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta}{\\cos \\theta}+\\frac{\\cos \\theta(1-\\sin \\theta)}{\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta}{\\cos \\theta}-\\frac{1-\\sin \\theta}{\\cos \\theta}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta+1-\\sin \\theta}{\\cos \\theta}$<\/p>\n\n\n\n

$\\frac{2}{\\cos \\theta}=2 \\sec \\theta$<\/p>\n\n\n\n

RHS<\/p>\n\n\n\n

Question 27<\/h4>\n\n\n\n

$\\frac{\\cos \\theta}{1-\\sin \\theta}+\\frac{\\cos \\theta}{1+\\sin \\theta}=\\frac{2}{\\cos \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{\\cos \\theta}{1+\\sin \\theta}+\\frac{\\cos \\theta}{1-\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{\\cos \\theta(1+\\sin \\theta)+\\cos \\theta(1-\\sin \\theta)}{(1-\\sin \\theta)(1+\\sin \\theta)}$<\/p>\n\n\n\n

$\\frac{\\cos \\theta+\\cos \\theta \\times \\sin \\theta+\\cos \\theta-\\cos \\theta \\times \\sin \\theta}{1^{2}-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{2 \\cos \\theta}{1-\\sin ^{2} \\theta}+\\frac{2 \\cos \\theta}{\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{2}{\\cos \\theta}$ proved<\/p>\n\n\n\n

Question 28<\/h4>\n\n\n\n

$\\frac{1}{1+\\cos \\theta}+\\frac{1}{1-\\cos \\theta}=\\frac{2}{\\sin ^{2} \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{1}{1+\\cos \\theta}+\\frac{1}{1-\\cos \\theta}$<\/p>\n\n\n\n

$\\frac{1-\\cos \\theta+1+\\cos \\theta}{(1+\\cos \\theta)(1-\\cos \\theta)}$<\/p>\n\n\n\n

$\\frac{2}{1^{2}-\\cos ^{2} \\theta}=\\frac{2}{1-\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{2}{\\sin ^{2} \\theta}$ proved<\/p>\n\n\n\n

Question 29<\/h4>\n\n\n\n

$\\frac{1}{1-\\sin \\theta}-\\frac{1}{1+\\sin \\theta}=\\frac{2 \\tan \\theta}{\\cos \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{1}{1-\\sin \\theta}+\\frac{1}{1+\\sin \\theta}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta-1-\\sin \\theta}{(1-\\sin \\theta)(1+\\sin \\theta)}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta-1+\\sin \\theta}{1^{2}-\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{2 \\sin \\theta}{1-\\sin ^{2} \\theta}=\\frac{2 \\sin \\theta}{\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{2 \\sin \\theta}{1-\\sin ^{2} \\theta} \\times \\frac{1}{\\cos \\theta}$<\/p>\n\n\n\n

$\\frac{2 \\tan \\theta}{1-\\frac{1}{\\cos \\theta}}$<\/p>\n\n\n\n

$\\frac{2 \\tan \\theta}{\\cos \\theta}$ = proved<\/p>\n\n\n\n

Question 30<\/h4>\n\n\n\n

$\\cot ^{2} \\theta-\\cos ^{2} \\theta=\\cot ^{2} \\theta \\cdot \\cos ^{2} \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\cot ^{2} \\theta-\\cos ^{2} \\theta$<\/p>\n\n\n\n

$\\frac{\\cos ^{2} \\theta}{\\sin ^{2} \\theta}-\\cos ^{2} \\theta$<\/p>\n\n\n\n

$\\cos ^{2} \\theta\\left[\\frac{1}{\\sin ^{2} \\theta}-1\\right]$<\/p>\n\n\n\n

$\\cos ^{2} \\theta\\left[\\frac{1-\\sin ^{2} \\theta}{\\sin ^{2} \\theta}\\right]$<\/p>\n\n\n\n

$\\cos ^{2} \\theta\\left[\\frac{\\cos ^{2} \\theta}{\\sin ^{2} \\theta}\\right]$<\/p>\n\n\n\n

$\\cos ^{2} \\theta \\times \\cot ^{2} \\theta$ proved<\/p>\n\n\n\n

Question 31<\/h4>\n\n\n\n

$\\tan ^{2} \\phi-\\sin ^{2} \\phi-\\tan ^{2} \\phi \\cdot \\sin ^{2} \\phi=0$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\tan ^{2} \\phi-\\sin ^{2} \\phi-\\tan ^{2} \\phi \\times \\sin ^{2} \\phi$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\phi}{\\cos ^{2} \\phi}-\\sin ^{2} \\phi-\\frac{\\sin ^{2} \\phi}{\\cos ^{2} \\phi} \\times \\sin ^{2} \\phi$<\/p>\n\n\n\n

$\\sin ^{2} \\phi\\left[\\frac{1}{\\cos ^{2} \\phi}-1\\right]-\\frac{\\sin ^{4} \\phi}{\\cos ^{2} \\phi}$<\/p>\n\n\n\n

$\\sin ^{2} \\phi\\left[\\frac{1-\\cos ^{2} \\phi}{\\cos ^{2} \\phi}\\right]-\\frac{\\sin ^{4} \\phi}{\\cos ^{2} \\phi}$<\/p>\n\n\n\n

$\\sin ^{2} \\phi\\left[\\frac{\\sin ^{2} \\phi}{\\cos ^{2} \\phi}\\right]-\\frac{\\sin ^{4} \\phi}{\\cos ^{2} \\phi}$<\/p>\n\n\n\n

$\\frac{\\sin ^{4} \\phi}{\\cos ^{2} \\phi}-\\frac{\\sin ^{4} \\phi}{\\cos ^{2} \\phi}$<\/p>\n\n\n\n

=0 proved<\/p>\n\n\n\n

Question 32<\/h4>\n\n\n\n

$\\tan ^{2} \\phi+\\cot ^{2} \\phi+2=\\sec ^{2} \\phi \\cdot \\operatorname{cosec}^{2} \\phi$<\/strong><\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\tan ^{2} \\phi+\\cot ^{2} \\phi+2$<\/p>\n\n\n\n

$\\sin ^{2} \\phi-1+\\operatorname{cosec}^{2} \\phi-1+2$<\/p>\n\n\n\n

$\\frac{1}{\\cos ^{2} \\phi}-1+\\frac{1}{\\sin ^{2} \\phi}-1+2$<\/p>\n\n\n\n

$\\frac{1}{\\cos ^{2} \\phi}+\\frac{1}{\\sin ^{2} \\phi}-\\not 2+\\not 2$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\phi+\\cos ^{2} \\phi}{\\cos ^{2} \\phi \\cdot \\sin ^{2} \\phi}=\\frac{1}{\\cos ^{2} \\phi \\cdot \\sin ^{2} \\phi}$<\/p>\n\n\n\n

$\\sin ^{2} \\phi \\times \\operatorname{cosec}^{2} \\phi$ proved<\/p>\n\n\n\n

Question 33<\/h4>\n\n\n\n

$\\frac{\\operatorname{cosec} \\theta+\\cot \\theta-1}{\\cot \\theta-\\operatorname{cosec} \\theta+1}=\\frac{1+\\cos \\theta}{\\sin \\theta}$<\/strong><\/p>\n\n\n\n

$\\frac{\\operatorname{cosec} \\theta+\\cot \\theta-1}{\\cot \\theta-\\operatorname{cosec} \\theta+1}$<\/p>\n\n\n\n

$\\frac{\\operatorname{cosec} \\theta+\\cot \\theta-\\left(\\operatorname{cosec}^{2} \\theta-\\cot ^{2} \\theta\\right)}{\\cot \\theta-\\operatorname{cosec} \\theta+1}$<\/p>\n\n\n\n

$\\frac{(\\operatorname{cosec} \\theta+\\cot \\theta)-(\\operatorname{cosec} \\theta-\\cot \\theta) \\times(\\operatorname{cosec} \\theta+\\cot \\theta)}{1-\\operatorname{cosec} \\theta+\\cot \\theta}$<\/p>\n\n\n\n

$\\frac{(\\operatorname{cosec} \\theta+\\cot \\theta) \\times[1-(\\operatorname{cosec} \\theta-\\cot \\theta)]}{1-\\operatorname{cosec} \\theta+\\cot \\theta}$<\/p>\n\n\n\n

$\\frac{(\\operatorname{cosec} \\theta+\\cot \\theta) \\times(1-\\operatorname{cosec} \\theta+\\cot \\theta)}{1-\\operatorname{cosec} \\theta+\\cot \\theta}$<\/p>\n\n\n\n

$=\\operatorname{cosec} \\theta+\\cot \\theta$<\/p>\n\n\n\n

$=\\frac{1}{\\sin \\theta}+\\frac{\\cos \\theta}{\\sin \\theta}$<\/p>\n\n\n\n

$=\\frac{1+\\cos \\theta}{\\sin \\theta}=$ proved<\/p>\n\n\n\n

Question 34<\/h4>\n\n\n\n

$\\frac{\\tan \\theta}{1-\\cot \\theta}+\\frac{\\cot \\theta}{1-\\tan \\theta}=1+\\tan \\theta+\\cot \\theta$<\/strong><\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{\\tan \\theta}{1-\\cot \\theta}+\\frac{\\cot \\theta}{1-\\tan \\theta}$<\/p>\n\n\n\n

$\\frac{\\frac{\\sin \\theta}{\\cos \\theta}}{1-\\frac{\\cos \\theta}{\\sin \\theta}}+\\frac{\\frac{\\cos \\theta}{\\sin \\theta}}{1-\\frac{\\sin \\theta}{\\cos \\theta}}$<\/p>\n\n\n\n

$\\frac{\\frac{\\sin \\theta}{\\cos \\theta}}{\\frac{\\sin \\theta-\\cos \\theta}{\\sin \\theta}}+\\frac{\\frac{\\cos \\theta}{\\sin \\theta}}{\\frac{\\cos \\theta-\\sin \\theta}{\\cos \\theta}}$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\theta}{\\cos \\theta(\\sin \\theta-\\cos \\theta)}+\\frac{\\cos ^{2} \\theta}{\\sin \\theta(\\cos \\theta-\\sin \\theta)}$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\theta}{\\cos \\theta(\\sin \\theta-\\cos \\theta)}-\\frac{\\cos ^{2} \\theta}{\\sin \\theta(\\sin \\theta-\\cos \\theta)}$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} \\theta-\\cos ^{2} \\theta}{\\cos \\theta \\sin \\theta(\\sin \\theta-\\cos \\theta)}$<\/p>\n\n\n\n

$\\frac{(\\sin \\theta-\\cos \\theta) \\times\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta+\\sin \\theta \\times \\cos \\theta\\right)}{\\cos \\theta \\times \\cot \\theta(\\sin \\theta-\\cos \\theta)}$<\/p>\n\n\n\n

$\\frac{\\sin \\theta \\times \\cos \\theta+\\sin ^{2} \\theta+\\cos ^{2} \\theta}{\\cos \\theta \\times \\sin \\theta}$<\/p>\n\n\n\n

$\\frac{\\sin \\theta \\times \\cos \\theta}{\\cos \\theta \\times \\sin \\theta}+\\frac{\\sin ^{2} \\theta}{\\cos \\theta \\times \\sin \\theta}+\\frac{\\cos ^{2} \\theta}{\\cos \\theta \\times \\sin \\theta}$<\/p>\n\n\n\n

$1+\\tan \\theta+\\cot \\theta$ proved<\/p>\n\n\n\n

Question 35<\/h4>\n\n\n\n

$\\frac{1-\\cos \\theta}{1+\\cos \\theta}=(\\cot \\theta-\\operatorname{cosec} \\theta)^{2}$<\/strong><\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\frac{1-\\cos \\theta}{1+\\cos \\theta}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\frac{1-\\cos \\theta}{1+\\cos \\theta} \\times \\frac{1-\\cos \\theta}{1-\\cos \\theta}$<\/p>\n\n\n\n

$\\frac{(1-\\cos \\theta)^{2}}{1^{2}-\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\frac{(1-\\cos \\theta)^{2}}{1-\\cos ^{2} \\theta}=\\frac{(1-\\cos \\theta)^{2}}{\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$\\left(\\frac{1-\\cos \\theta}{\\sin \\theta}\\right)^{2}$<\/p>\n\n\n\n

$\\left(\\frac{1-\\cos \\theta}{\\sin \\theta}-\\frac{\\cos \\theta}{\\sin \\theta}\\right)^{2}$<\/p>\n\n\n\n

$=(\\cos \\theta-\\operatorname{cosec} \\theta)^{2}$<\/p>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u0915\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 :<\/strong><\/p>\n\n\n\n

Question 36<\/h4>\n\n\n\n

$\\sqrt{\\frac{1-\\cos \\theta}{1+\\cos \\theta}}=\\frac{1-\\cos \\theta}{\\sin \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sqrt{\\frac{1+\\cos \\theta}{1-\\cos \\theta}}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\sqrt{\\frac{1+\\cos \\theta}{1-\\cos \\theta} \\times \\frac{1+\\cos \\theta}{1+\\cos \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{(1+\\cos \\theta)^{2}}{1^{2}-\\cos ^{2} \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{(1+\\cos \\theta)^{2}}{1-\\cos ^{2} \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{(1+\\cos \\theta)^{2}}{\\sin ^{2} \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\left(\\frac{1+\\cos \\theta}{\\sin \\theta}\\right)^{2}}$<\/p>\n\n\n\n

$\\frac{1+\\cos \\theta}{\\sin \\theta}$=Proved<\/p>\n\n\n\n

Question 37<\/h4>\n\n\n\n

$\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta}}=\\frac{\\cos \\theta}{1+\\sin \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sqrt{\\frac{1-\\cos \\theta}{1+\\cos \\theta}}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\sqrt{\\frac{1-\\cos \\theta}{1+\\cos \\theta} \\times \\frac{1-\\cos \\theta}{1-\\cos \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{(1-\\cos \\theta)^{2}}{1^{2}-\\cos ^{2} \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{(1-\\cos \\theta)^{2}}{1-\\cos ^{2} \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{(1-\\cos \\theta)^{2}}{\\sin ^{2} \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\left(\\frac{1-\\cos \\theta}{\\sin \\theta}\\right)^{2}}$<\/p>\n\n\n\n

$\\frac{1-\\cos \\theta}{\\sin \\theta}=$ proved<\/p>\n\n\n\n

Question 38<\/h4>\n\n\n\n

$\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta}}=\\sec \\theta-\\tan \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta}}$<\/p>\n\n\n\n

\u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta}} \\times \\frac{1-\\sin \\theta}{1-\\sin \\theta}$<\/p>\n\n\n\n

$\\sqrt{\\frac{(1-\\sin \\theta)^{2}}{1^{2}+\\sin ^{2} \\theta}}=\\sqrt{\\frac{(1-\\sin \\theta)^{2}}{1+\\sin ^{2} \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{(1-\\sin \\theta)^{2}}{\\cos ^{2} \\theta}}=\\sqrt{\\left(\\frac{1-\\sin \\theta}{\\cos \\theta}\\right)^{2}}$<\/p>\n\n\n\n

$\\frac{1-\\sin \\theta}{\\cos \\theta}=\\frac{1}{\\cos \\theta}-\\frac{\\sin \\theta}{\\cos \\theta}$<\/p>\n\n\n\n

$\\sec \\theta-\\tan \\theta$=RHS<\/p>\n\n\n\n

Question 39<\/h4>\n\n\n\n

$\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta}}=\\frac{\\cos \\theta}{1+\\sin \\theta}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta}}$<\/p>\n\n\n\n

\u0905\u0902\u0936 \u0915\u093e \u092a\u0930\u093f\u092e\u0947\u092f\u0915\u0930\u0923 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta} \\times \\frac{1+\\sin \\theta}{1+\\sin \\theta}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{1^{2}-\\sin ^{2} \\theta}{(1+\\sin \\theta)^{2}}}=\\sqrt{\\frac{1-\\sin ^{2} \\theta}{(1+\\sin \\theta)^{2}}}$<\/p>\n\n\n\n

$\\sqrt{\\frac{\\cos ^{2} \\theta}{(1+\\sin \\theta)^{2}}}=\\sqrt{\\left(\\frac{\\cos \\theta}{1+\\sin \\theta}\\right)^{2}}$<\/p>\n\n\n\n

$\\frac{\\cos \\theta}{1+\\sin \\theta}=\\text { R.H.S. }$<\/p>\n\n\n\n

Question 40<\/h4>\n\n\n\n

$\\sqrt{\\frac{1+\\sin \\theta}{1-\\sin \\theta}}+\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta}}=2 \\sec \\theta$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sqrt{\\frac{1+\\sin \\theta}{1-\\sin \\theta}}+\\sqrt{\\frac{1-\\sin \\theta}{1+\\sin \\theta}}$<\/p>\n\n\n\n

$\\frac{\\sqrt{1+\\sin \\theta}}{\\sqrt{1-\\sin \\theta}}+\\frac{\\sqrt{1-\\sin \\theta}}{\\sqrt{1+\\sin \\theta}}$<\/p>\n\n\n\n

$\\frac{(\\sqrt{1+\\sin \\theta})^{2}+(\\sqrt{1-\\sin \\theta})^{2}}{(\\sqrt{1-\\sin \\theta})(\\sqrt{1+\\sin \\theta})}$<\/p>\n\n\n\n

$\\frac{(1+\\sin \\theta)+(1-\\sin \\theta)}{(\\sqrt{1-\\sin \\theta})(\\sqrt{1+\\sin \\theta})}$<\/p>\n\n\n\n

$\\frac{1+\\sin \\theta+1-\\sin \\theta}{\\sqrt{1^{2}-\\sin ^{2} \\theta}}$<\/p>\n\n\n\n

$\\frac{2}{\\sqrt{1-\\sin ^{2} \\theta}}=\\frac{2}{\\cos \\theta}$<\/p>\n\n\n\n

$2 \\times \\frac{1}{\\cos \\theta}=2 \\sec \\theta$=proved<\/p>\n\n\n\n

Type IV : \u0936\u0902\u0930\u094d\u0924 \u0935\u093e\u0932\u0947 \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u0913\u0902 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 :<\/strong><\/p>\n\n\n\n

Question 41<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\sec \\theta+\\tan \\theta=m$ \u0914\u0930 $\\sec \\theta-\\tan \\theta=n$, \u0924\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $\\sqrt{m n}=1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\sqrt{m n}=\\sqrt{\\sec \\theta+\\tan \\theta(\\sec \\theta-\\tan \\theta)}$<\/p>\n\n\n\n

$\\sqrt{\\sec ^{2} \\theta-\\sec \\theta \\times \\tan \\theta+\\sec \\theta \\times \\tan \\theta-\\tan ^{2} \\theta}$<\/p>\n\n\n\n

$\\sqrt{\\sec ^{2} \\theta-\\tan ^{2} \\theta}=\\sqrt{1}$<\/p>\n\n\n\n

=1 proved<\/p>\n\n\n\n

Question 42<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\cos \\theta+\\sin \\theta=1$, \u0924\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $\\cos \\theta-\\sin \\theta=\\pm 1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e $-\\cos \\theta-\\sin \\theta=\\mathrm{x}$<\/p>\n\n\n\n

\u0926\u093f\u092f\u093e \u0939\u0948 – $\\cos \\theta+\\sin \\theta=1$<\/p>\n\n\n\n

\u0926\u094b\u0928\u094b\u0902 \u0915\u094b \u0935\u0930\u094d\u0917 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$(\\cos \\theta-\\sin \\theta)^{2}+(\\cos \\theta+\\sin \\theta)^{2}=x^{2}+1^{2}$<\/p>\n\n\n\n

$\\cos ^{2} \\theta+\\sin ^{2} \\theta-2 \\cos \\theta \\cdot \\sin \\theta+\\cos ^{2} \\theta+\\sin ^{2} \\theta+2 \\cos \\theta \\cdot \\sin \\theta=x^{2}+1^{2}$<\/p>\n\n\n\n

$\\cos ^{2} \\theta+\\sin ^{2} \\theta+\\cos ^{2} \\theta+\\sin ^{2} \\theta=x^{2}+1$<\/p>\n\n\n\n

$1+1=x^{2}+1$<\/p>\n\n\n\n

$2=x^{2}+1$<\/p>\n\n\n\n

$x^{2}+1=2$<\/p>\n\n\n\n

$x^{2}=2-1$<\/p>\n\n\n\n

$x=\\sqrt{1}$<\/p>\n\n\n\n

$x=\\pm 1$<\/p>\n\n\n\n

$\\cos \\theta+\\sin \\theta=x$<\/p>\n\n\n\n

$\\cos \\theta+\\sin \\theta=\\pm 1$<\/p>\n\n\n\n

Question 43<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\sin \\theta+\\sin ^{2} \\theta=1$. \u0924\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $\\cos ^{2} \\theta+\\cos ^{4} \\theta=1$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sin \\theta+\\sin ^{2} \\theta=1$<\/p>\n\n\n\n

$\\sin \\theta=1-\\sin ^{2} \\theta$<\/p>\n\n\n\n

$\\sin \\theta+\\cos ^{2} \\theta$<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\cos ^{2} \\theta+\\cos ^{4} \\theta$<\/p>\n\n\n\n

$\\cos ^{2} \\theta+\\left(\\cos ^{2} \\theta\\right)^{2}$<\/p>\n\n\n\n

$\\cos ^{2} \\theta+\\sin ^{2} \\theta$<\/p>\n\n\n\n

=1 proved<\/p>\n\n\n\n

Question 44<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\tan \\theta+\\sec \\theta=x$. \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $\\sin \\theta=\\frac{x^{2}-1}{x^{2}+1}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sec \\theta+\\tan \\theta=\\mathrm{x}$….(i)<\/p>\n\n\n\n

\u0939\u092e \u091c\u093e\u0928\u0924\u0947 \u0939\u0948 \u0915\u093f $\\sec ^{2} \\theta-\\tan ^{2} \\theta=1$<\/p>\n\n\n\n

$(\\sec \\theta-\\tan \\theta)(\\sec \\theta+\\tan \\theta)=1$<\/p>\n\n\n\n

$(\\sec \\theta-\\tan \\theta) \\cdot x=1$<\/p>\n\n\n\n

$\\sec \\theta-\\tan \\theta=\\frac{1}{x}$…(ii)<\/p>\n\n\n\n

\u0938\u093e\u092e\u0940\u0915\u0930\u0923 (i) \u0924\u0925\u093e (ii) \u0938\u0947<\/p>\n\n\n\n

$\\sec \\theta+\\tan \\theta=x$<\/p>\n\n\n\n

$\\sec \\theta-\\tan \\theta=\\frac{1}{x}$<\/p>\n\n\n\n

$2 \\sec \\theta=x+\\frac{1}{x}$<\/p>\n\n\n\n

$2 \\sec \\theta=\\frac{x^{2}+1}{x}$<\/p>\n\n\n\n

$\\sec \\theta=\\frac{x^{2}+1}{2 x}=\\frac{\\mathrm{AC}}{\\mathrm{BC}}$<\/p>\n\n\n\n

\"\"\/<\/a><\/figure>\n\n\n\n

$\\mathrm{AB}=\\sqrt{(\\mathrm{AC})^{2}-(\\mathrm{BC})^{2}}$<\/p>\n\n\n\n

$\\mathrm{AB}=\\sqrt{\\left(x^{2}+1\\right)^{2}-(2 \\mathrm{x})^{2}}$<\/p>\n\n\n\n

$\\mathrm{AB}=\\sqrt{\\left(\\left(x^{2}\\right)^{2}+(1)^{2}+2 \\cdot x^{2} \\cdot 1\\right)-4 x^{2}}$<\/p>\n\n\n\n

$A B=\\sqrt{\\left(x^{2}\\right)^{2}+1+2 x^{2}-4 x^{2}}$<\/p>\n\n\n\n

$A B=\\sqrt{\\left(x^{2}\\right)^{2}-2 x^{2}+1}$<\/p>\n\n\n\n

$A B=\\sqrt{\\left(x^{2}-1\\right)^{2}}$<\/p>\n\n\n\n

$A B=x^{2}-1$<\/p>\n\n\n\n

$\\sin \\theta=\\frac{A B}{A C}=\\frac{x^{2}-1}{x^{2}+1}$<\/p>\n\n\n\n

$\\sin \\theta=\\frac{x^{2}-1}{x^{2}+1}$<\/p>\n\n\n\n

Question 45<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\sin \\theta+\\cos \\theta=p$ \u0914\u0930 $\\sec \\theta+\\operatorname{cosec} \\theta=q$, \u0924\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $q\\left(p^{2}-1\\right)=2 p$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$\\mathrm{q}\\left(\\mathrm{p}^{2}-1\\right)$<\/p>\n\n\n\n

$(\\sec \\theta+\\operatorname{cosec} \\theta)\\left[(\\sin \\theta+\\cos \\theta)^{2}-1\\right]$<\/p>\n\n\n\n

$\\frac{1}{\\cos \\theta}+\\frac{1}{\\sin \\theta}\\left[\\sin ^{2} \\theta+\\cos \\theta^{2}+2 \\sin \\theta \\cdot \\cos \\theta-1\\right]$<\/p>\n\n\n\n

$\\frac{\\sin \\theta+\\cos \\theta}{\\cos \\theta \\cdot \\sin \\theta}[1+2 \\sin \\theta \\cdot \\cos \\theta- 1]$<\/p>\n\n\n\n

$\\frac{\\sin \\theta+\\cos \\theta}{\\cos \\theta \\cdot \\sin \\theta} \\times 2 \\sin \\theta \\cdot \\cos \\theta$<\/p>\n\n\n\n

=2p proved<\/p>\n\n\n\n

Question 46<\/h4>\n\n\n\n

\u092f\u0926\u093f $x \\cos \\theta=a$ \u0914\u0930 $y=a \\tan \\theta$, \u0924\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $x^{2}-y^{2}=a^{2}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$x \\cos \\theta=a$<\/p>\n\n\n\n

$x=\\frac{a}{\\cos \\theta}$<\/p>\n\n\n\n

$x=a \\sec \\theta, \\quad y=a \\tan \\theta$<\/p>\n\n\n\n

\u0926\u094b\u0928\u094b\u0902 \u0915\u094b \u0935\u0930\u094d\u0917 \u0915\u0930\u0915\u0947 \u0918\u091f\u093e\u0928\u0947 \u092a\u0930 <\/p>\n\n\n\n

$x^{2}-y^{2}=(a \\sec \\theta)^{2}-(a \\tan \\theta)^{2}$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2} \\sec ^{2} \\theta-a^{2} \\tan ^{2} \\theta$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2}\\left(\\sec ^{2} \\theta-\\tan ^{2} \\theta\\right)$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2}(1)^{2}$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2}$<\/p>\n\n\n\n

Question 47 (to be fixed)<\/h4>\n\n\n\n

\u092f\u0926\u093f $x=r \\cos \\alpha$. $\\sin \\beta, y=r \\cos \\alpha \\cdot \\cos \\beta, z=r \\sin \\alpha$, \u0924\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $x^{2}+y^{2}+z^{2}=r$ ?<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

LHS<\/p>\n\n\n\n

$x^{2}+y^{2}+z^{2}=(r \\cos \\alpha \\cdot \\sin \\beta)^{2}+(r \\cos \\alpha \\cdot \\cos \\beta)^{2}+(r \\sin \\alpha)^{2}$<\/p>\n\n\n\n

$x^{2}+y^{2}+z^{2}=r^{2} \\cos ^{2} \\alpha \\cdot \\sin ^{2} \\beta+r^{2} \\cos ^{2} \\alpha \\cdot \\cos ^{2} \\beta+r \\sin ^{2} \\alpha$<\/p>\n\n\n\n

$x^{2}+y^{2}+z^{2}=r^{2} \\cos ^{2} \\alpha\\left(\\sin ^{2} \\beta+\\cos ^{2} \\beta\\right)+r^{2} \\sin ^{2} \\alpha$<\/p>\n\n\n\n

$x^{2}+y^{2}+z^{2}=r^{2} \\cos ^{2} \\alpha(1)+r^{2} \\cdot \\sin ^{2} \\alpha$<\/p>\n\n\n\n

$x^{2}+y^{2}+z^{2}=r^{2} \\cos ^{2} \\alpha+r^{2} \\cdot \\sin ^{2} \\alpha$<\/p>\n\n\n\n

$x^{2}+y^{2}+z^{2}=r^{2}\\left(\\cos ^{2} \\alpha+\\sin ^{2} \\alpha\\right)$<\/p>\n\n\n\n

$x^{2}+y^{2}+z^{2}=r^{2}(1)^{2}$<\/p>\n\n\n\n

$x^{2}+y^{2}+z^{2}=r^{2}=$ R.H.S.<\/p>\n\n\n\n

Question 48<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\sec \\theta-\\tan \\theta=x$, \u0924\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f<\/strong><\/p>\n\n\n\n

(i) $\\cos \\theta=\\frac{2 x}{1+x^{2}}$<\/strong><\/p>\n\n\n\n

(ii) $\\sin \\theta=\\frac{1-x^{2}}{1+x^{2}}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\sec \\theta-\\tan \\theta=\\mathrm{x}$…(i)<\/p>\n\n\n\n

\u0939\u092e \u091c\u093e\u0928\u0924\u0947 \u0939\u0948 \u0915\u093f $\\left(\\sec ^{2} \\theta-\\tan ^{2} \\theta\\right)=1$<\/p>\n\n\n\n

$(\\sec \\theta+\\tan \\theta)(\\sec \\theta-\\tan \\theta)=1$<\/p>\n\n\n\n

$(\\sec \\theta+\\tan \\theta) \\cdot \\mathrm{x}=1$<\/p>\n\n\n\n

$\\sec \\theta+\\tan \\theta=\\frac{1}{x}-$ (ii)<\/p>\n\n\n\n

\u0938\u093e\u092e\u0940\u0915\u0930\u0923 (i) \u0924\u0925\u093e (ii) \u0938\u0947<\/p>\n\n\n\n

$\\sec \\theta-\\tan \\theta=x$<\/p>\n\n\n\n

$\\sec \\theta+\\tan \\theta=\\frac{1}{x}$<\/p>\n\n\n\n

\"\"\/<\/a><\/figure>\n\n\n\n

$2 \\sec \\theta=x+\\frac{1}{x}$<\/p>\n\n\n\n

$2 \\sec \\theta=\\frac{x^{2}+1}{x}$<\/p>\n\n\n\n

$\\sec \\theta=\\frac{x^{2}+1}{2 x}=\\frac{\\mathrm{AC}}{\\mathrm{BC}}$<\/p>\n\n\n\n

$A B=\\sqrt{(A C)^{2}-(B C)^{2}}$<\/p>\n\n\n\n

$A B=\\sqrt{\\left(x^{2}+1\\right)^{2}-(2 x)^{2}}$<\/p>\n\n\n\n

$A B=\\sqrt{\\left(x^{2}\\right)^{2}+(1)^{2}+2 x^{2} \\cdot 1-4 x^{2}}$<\/p>\n\n\n\n

$A B=\\sqrt{\\left(x^{2}\\right)^{2}+1+2 x^{2}-4 x^{2}}$<\/p>\n\n\n\n

$A B=\\sqrt{\\left(x^{2}\\right)^{2}+1-2 x^{2}}$<\/p>\n\n\n\n

$A B=\\sqrt{\\left(x^{2}-1\\right)^{2}}$<\/p>\n\n\n\n

$A B=x^{2}-1$<\/p>\n\n\n\n

$\\cos \\theta=\\frac{\\mathrm{BC}}{\\mathrm{AC}}=\\frac{2 x}{x^{2}+1}$<\/p>\n\n\n\n

$\\sin \\theta=\\frac{\\mathrm{AB}}{\\mathrm{AC}}=\\frac{x^{2}-1}{x^{2}+1}$<\/p>\n\n\n\n

Question 49<\/h4>\n\n\n\n

\u092f\u0926\u093f $a \\cos \\theta+b \\sin \\theta=c$, \u0924\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $a \\sin \\theta-b \\cos \\theta=\\pm \\sqrt{a^{2}+b^{2}-c^{2}}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

\u092e\u093e\u0928\u093e $-\\mathrm{a} \\sin \\theta-\\mathrm{b} \\cos \\theta=\\mathrm{x}$…(i)<\/p>\n\n\n\n

\u0926\u093f\u092f\u093e \u0939\u0948 – $\\mathrm{a} \\cos \\theta+\\mathrm{b} \\sin \\theta=\\mathrm{c}$…(ii)<\/p>\n\n\n\n

\u0926\u094b\u0928\u094b\u0902 \u0915\u094b \u0935\u0930\u094d\u0917 \u0915\u0930\u0915\u0947 \u091c\u094b\u0921\u093c\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$(a \\sin \\theta-b \\cos \\theta)^{2}+(a \\cos \\theta+b \\sin \\theta)^{2}=x^{2}+c^{2}$<\/p>\n\n\n\n

$a^{2} \\sin ^{2} \\theta+b^{2} \\cos ^{2} \\theta-2 a \\sin \\theta \\cdot b \\cos \\theta+a^{2} \\cos ^{2} \\theta+b^{2} \\sin ^{2} \\theta+$ $2 a \\cos \\theta \\cdot b \\sin \\theta=x^{2}+c^{2}$<\/p>\n\n\n\n

$a^{2} \\sin ^{2} \\theta+b^{2} \\cos ^{2} \\theta+a^{2} \\cos ^{2} \\theta+b^{2} \\sin ^{2} \\theta=x^{2}+c^{2}$<\/p>\n\n\n\n

$\\left(a^{2} \\sin ^{2} \\theta+a^{2} \\cos ^{2} \\theta\\right)+\\left(b^{2} \\cos ^{2} \\theta+b^{2} \\sin ^{2} \\theta\\right)=x^{2}+c^{2}$<\/p>\n\n\n\n

$a^{2}\\left(\\sin ^{2} \\theta+\\cos ^{2} \\theta\\right)+b^{2}\\left(\\cos ^{2} \\theta+\\sin ^{2} \\theta\\right)=x^{2}+c^{2}$<\/p>\n\n\n\n

$a^{2}=1+b^{2} \\times 1=x^{2}+c^{2}$<\/p>\n\n\n\n

$a^{2}+b^{2}=x^{2}+c^{2}$<\/p>\n\n\n\n

$x^{2}+c^{2}=a^{2}+b^{2}$<\/p>\n\n\n\n

$x^{2}=a^{2}+b^{2}-c^{2}$<\/p>\n\n\n\n

$x=\\pm \\sqrt{a^{2}+b^{2}-c^{2}}$<\/p>\n\n\n\n

$x=\\pm \\sqrt{a^{2}+b^{2}-c^{2}}$<\/p>\n\n\n\n

Question 50<\/h4>\n\n\n\n

\u092f\u0926\u093f $1+\\sin ^{2} \\theta=3 \\sin \\theta \\cdot \\cos \\theta$, \u0924\u092c \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $\\tan \\theta=1$ \u092f\u093e, $\\frac{1}{2}$, \u091c\u0939\u093e\u0901 $\\theta<90^{\\circ}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$1+\\sin ^{2} \\theta-3 \\sin \\theta \\cos \\theta$<\/p>\n\n\n\n

\u0926\u094b\u0928\u094b\u0902 \u0924\u0930\u092b $\\cos ^{2} \\theta$ \u0938\u0947 \u092d\u093e\u0917 \u0926\u0947\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\frac{1}{\\cos ^{2} \\theta}+\\frac{\\sin ^{2} \\theta}{\\cos ^{2} \\theta}=\\frac{3 \\cdot \\sin \\theta \\cos \\theta}{\\cos ^{2} \\theta}$<\/p>\n\n\n\n

$\\sec ^{2} \\theta+\\tan ^{2} \\theta=3 \\tan \\theta$<\/p>\n\n\n\n

$1+\\tan ^{2} \\theta+\\tan ^{2} \\theta-3 \\tan \\theta=0$<\/p>\n\n\n\n

$1+2 \\tan ^{2} \\theta-3 \\tan \\theta=0$<\/p>\n\n\n\n

$2 \\tan ^{2} \\theta-3 \\tan \\theta+1=0$<\/p>\n\n\n\n

$2 \\tan ^{2} \\theta-2 \\tan \\theta-\\tan \\theta+1=0$<\/p>\n\n\n\n

$2 \\tan \\theta(\\tan \\theta-1)-1(\\tan \\theta-1)=0$<\/p>\n\n\n\n

$\\begin{array}{r|rl}(\\tan \\theta-1) & 2 \\tan \\theta & -1 \\\\\\tan \\theta=1 & 2 \\tan \\theta & =1 \\\\&\\tan \\theta &=\\frac{1}{2}\\end{array}$<\/p>\n\n\n\n

$\\tan \\theta=1$ \u092f\u093e $\\frac{1}{2}$<\/p>\n\n\n\n

Question 51<\/h4>\n\n\n\n

\u092f\u0926\u093f $a \\cos \\theta-b \\sin \\theta=x$ \u0914\u0930 $a \\sin \\theta+b \\cos \\theta=y$, \u0924\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $a^{2}+b^{2}=x^{2}+y^{2}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$a \\cos \\theta-b \\sin \\theta=x$….(i)<\/p>\n\n\n\n

$a \\sin \\theta+b \\cos \\theta=y$…..(ii)<\/p>\n\n\n\n

\u0926\u094b\u0928\u094b\u0902 \u0938\u092e\u0940\u0915\u0930\u0923 \u0915\u094b \u0935\u0930\u094d\u0917 \u0915\u0930\u0915\u0947 \u091c\u094b\u0921\u093c\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$(a \\cos \\theta-b \\sin \\theta)^{2}+(a \\sin \\theta+b \\cos \\theta)=x^{2}+y^{2}$<\/p>\n\n\n\n

$a^{2} \\cos ^{2} \\theta+b^{2} \\sin ^{2} \\theta-2 a \\cos \\theta \\cdot b \\sin \\theta+a^{2} \\sin ^{2} \\theta+b^{2} \\cos ^{2} \\theta+2 a \\sin \\theta \\cdot b \\cos \\theta=x^{2}+y^{2}$<\/p>\n\n\n\n

$a^{2} \\cos ^{2} \\theta+b^{2} \\sin ^{2} \\theta+a^{2} \\sin ^{2} \\theta+b^{2} \\cos ^{2} \\theta=x^{2}+y^{2}$<\/p>\n\n\n\n

$\\left(a^{2} \\cos ^{2} \\theta+a^{2} \\sin ^{2} \\theta\\right)+\\left(b^{2} \\sin ^{2} \\theta+b^{2} \\cos ^{2} \\theta\\right)=x^{2}+y^{2}$<\/p>\n\n\n\n

$a^{2} \\times 1+b^{2} \\times 1=x^{2}+y^{2}$<\/p>\n\n\n\n

$a^{2}+b^{2}=x^{2}+y^{2}$<\/p>\n\n\n\n

Question 52<\/h4>\n\n\n\n

\u092f\u0926\u093f $x=a \\sec \\theta+b \\tan \\theta$ \u0914\u0930 $y=a \\tan \\theta+b \\sec \\theta$, \u0924\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $x^{2}-y^{2}=a^{2}-b^{2}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$x=a \\sec \\theta+b \\tan \\theta$…(i)<\/p>\n\n\n\n

$y=a \\tan \\theta+b \\sec \\theta$…(ii)<\/p>\n\n\n\n

\u0938\u092e\u0940\u0915\u0930\u0923 (i) \u0924\u0925\u093e (ii) \u0915\u094b \u0935\u0930\u094d\u0917 \u0915\u0930\u0915\u0947 \u0918\u091f\u093e\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$x^{2}-y^{2}=(a \\sec \\theta-b \\tan \\theta)^{2}-(a \\tan \\theta+b \\sec \\theta)^{2}$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2} \\sec ^{2} \\theta+b^{2} \\tan ^{2} \\theta+2 a \\sec \\theta \\cdot b \\tan \\theta-\\left(a^{2} \\tan ^{2} \\theta+b^{2} \\sec ^{2} \\theta+\\right. 2 \\operatorname{atan} \\theta \\cdot b \\sec \\theta)$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2} \\sec ^{2} \\theta+b^{2} \\tan ^{2} \\theta+2 a \\sec \\theta \\cdot b \\tan \\theta-a^{2} \\tan ^{2} \\theta-b^{2} \\sec ^{2} \\theta-2 a \\tan \\theta \\cdot b \\sec \\theta$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2} \\sec ^{2} \\theta+b^{2} \\tan ^{2} \\theta-a^{2} \\tan ^{2} \\theta-b^{2} \\sec ^{2} \\theta$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2} \\sec ^{2} \\theta-a^{2} \\tan ^{2} \\theta+b^{2} \\tan ^{2} \\theta-b^{2} \\sec ^{2} \\theta$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2}\\left(\\sec ^{2} \\theta-\\tan ^{2} \\theta\\right)-b^{2}\\left(\\tan ^{2} \\theta-\\sec ^{2} \\theta\\right)$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2}\\left(\\sec ^{2} \\theta-\\tan ^{2} \\theta\\right)-b^{2}\\left(\\sec ^{2} \\theta-\\tan ^{2} \\theta\\right)$<\/p>\n\n\n\n

$x^{2}-y^{2}=a^{2}-b^{2}$<\/p>\n\n\n\n

Question 53<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\left(a^{2}-b^{2}\\right) \\sin \\theta+2 a b \\cos \\theta=a^{2}+b^{2}$, \u0924\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0930\u0947\u0902 \u0915\u093f $\\tan \\theta=\\frac{a^{2}-b^{2}}{2 a b}$<\/strong><\/p>\n\n\n\n

Sol :<\/p>\n\n\n\n

$\\left(a^{2}-b^{2}\\right) \\sin \\theta+2 a b \\cos \\theta=a^{2}+b^{2}$<\/p>\n\n\n\n

\u0926\u094b\u0928\u094b\u0902 \u0924\u0930\u092b $\\cos \\theta$ \u0938\u0947 \u092d\u093e\u0917 \u0926\u0947\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\left(a^{2}-b^{2}\\right) \\times \\frac{\\sin \\theta}{\\cos \\theta}+(2 a b) \\times \\frac{\\cos \\theta}{\\cos \\theta}=\\frac{a^{2}}{\\cos \\theta}$<\/p>\n\n\n\n

$a^{2}-b^{2} \\tan \\theta+2 a b=a^{2} \\sec \\theta+b^{2} \\sec \\theta$<\/p>\n\n\n\n

$a^{2}-b^{2} \\tan \\theta+2 \\mathrm{ab}=a^{2}+b^{2}(\\sec \\theta)$<\/p>\n\n\n\n

$a^{2}-b^{2} \\tan \\theta+2 \\mathrm{ab}=\\left(a^{2}+b^{2}\\right) \\sqrt{1+\\tan ^{2} \\theta}$<\/p>\n\n\n\n

\u0926\u094b\u0928\u094b\u0902 \u0924\u0930\u092b \u0935\u0930\u094d\u0917 \u0915\u0930\u0928\u0947 \u092a\u0930<\/p>\n\n\n\n

$\\left[\\left(a^{2}-b^{2}\\right) \\tan \\theta+2 a b\\right]^{2}=\\left[\\left(a^{2}-b^{2}\\right) \\sqrt{1+\\tan ^{2}} \\theta\\right]^{2}$<\/p>\n\n\n\n

$\\left(a^{2}-b^{2} \\tan \\theta\\right)^{2}+(2 a b)^{2}+2\\left(a^{2}-b^{2}\\right) \\tan \\theta \\cdot 2 a b=\\left(a^{2}+b^{2}\\right)^{2}+\\left(1+\\tan ^{2} \\theta\\right)$<\/p>\n\n\n\n

$\\left(a^{2}-b^{2}\\right) \\times \\tan ^{2} \\theta+4 a b\\left(a^{2}-b^{2}\\right) \\times \\tan \\theta+4 a^{2} b^{2}=\\left(a^{2}+b^{2}\\right)^{2}+\\left(a^{2}+b^{2}\\right)^{2} \\times \\tan ^{2} \\theta$<\/p>\n\n\n\n

$\\left(a^{2}-b^{2}\\right)^{2} \\times \\tan ^{2} \\theta-\\left(a^{2}-b^{2}\\right)^{2} \\tan ^{2} \\theta+4 a b\\left(a^{2}-b^{2}\\right) \\tan \\theta+4 a^{2} b^{2}-\\left(a^{2}+b^{2}\\right)^{2}=0$<\/p>\n\n\n\n

$\\left[a^{4}-2 a^{2} \\cdot b^{2}+b^{4}-a^{4}-2 a^{2} \\cdot b^{2}-b^{4}\\right] \\tan ^{2} \\theta+4 a b\\left(a^{2}-b^{2}\\right)\\tan \\theta+4 a^{2} \\cdot b^{2}-a^{4}-b^{4} 2 a^{2} b^{2}=0$<\/p>\n\n\n\n

$-4 a^{2} \\cdot b^{2} \\times \\tan ^{2} \\theta+4 a b\\left(a^{2}-b^{2}\\right) \\tan \\theta-a^{2}-b^{4}+2 a^{2} \\cdot b^{2}=0$<\/p>\n\n\n\n

$4 a^{2} \\cdot b^{2} \\times \\tan ^{2} \\theta-4 a b\\left(a^{2}-b^{2}\\right) \\tan \\theta+a^{4}+b^{4}-2 a^{2} \\cdot b^{2}=0$<\/p>\n\n\n\n

$4 a^{2} \\cdot b^{2} \\times \\tan ^{2} \\theta-4 a b\\left(a^{2}-b^{2}\\right) \\tan\\theta+\\left(a^{2}\\right)^{2}+\\left(b^{2}\\right)^{2}-2 a^{2} \\cdot b^{2}=0$<\/p>\n\n\n\n

$(2 a b \\tan \\theta)^{2}-2 \\cdot 2 a b \\tan \\theta\\left(a^{2}-b^{2}\\right)+\\left(a^{2}-b^{2}\\right)^{2}=0$<\/p>\n\n\n\n

$\\left[2 a b \\tan \\theta-\\left(a^{2}-b^{2}\\right)\\right]^{2}=0$<\/p>\n\n\n\n

$2 a b \\tan \\theta-\\left(a^{2}-b^{2}\\right)=0$<\/p>\n\n\n\n

$2 a b \\tan \\theta=a^{2}-b^{2}$<\/p>\n\n\n\n

$\\tan \\theta=\\frac{a^{2}-b^{2}}{2 a b}$<\/p>\n\n\n\n

\n
KC Sinha Solutions<\/a><\/div>\n\n\n\n
KC Sinha Class 10 Solutions<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Type-I : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u090f\u0935\u0902 \u092c\u0940\u091c\u0940\u092f \u0938\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u0940\u0927\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 : Question 1 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0930\u0947\u0902 : (i) $\\sin ^{2} \\theta+\\cos ^{2} \\theta=\\ldots \\ldots \\ldots .$ Sol : $\\sin ^{2} \\theta+\\cos ^{2} \\theta=1$ (ii) $1+\\tan ^{2} \\theta=$…. Sol : $1+\\tan ^{2} \\theta=\\sec ^{2} \\theta$ (iii) $\\sin \\theta \\cdot \\cot \\theta$ \u0915\u093e […]<\/p>\n","protected":false},"author":302,"featured_media":625677,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"newspack_featured_image_position":"","newspack_post_subtitle":"","newspack_article_summary_title":"Overview:","newspack_article_summary":"","newspack_hide_updated_date":false,"newspack_show_updated_date":false,"footnotes":""},"categories":[24],"tags":[],"boards":[],"class_list":["post-625698","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-10","entry"],"yoast_head":"\nKC Sinha: Exercise 8.4 - Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f - IndCareer Schools<\/title>\n<meta name=\"description\" content=\"Type-I : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u090f\u0935\u0902 \u092c\u0940\u091c\u0940\u092f \u0938\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u0940\u0927\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 : Question 1 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0930\u0947\u0902 : (i) $sin ^{2} theta+cos ^{2}\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"KC Sinha: Exercise 8.4 - Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f\" \/>\n<meta property=\"og:description\" content=\"Type-I : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u090f\u0935\u0902 \u092c\u0940\u091c\u0940\u092f \u0938\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u0940\u0927\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 : Question 1 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0930\u0947\u0902 : (i) $sin ^{2} theta+cos ^{2}\" \/>\n<meta property=\"og:url\" content=\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/\" \/>\n<meta property=\"og:site_name\" content=\"IndCareer Schools\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/indcareer\" \/>\n<meta property=\"article:published_time\" content=\"2023-09-08T01:41:32+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-09-08T01:44:19+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg\" \/>\n\t<meta property=\"og:image:width\" content=\"1600\" \/>\n\t<meta property=\"og:image:height\" content=\"901\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\n<meta name=\"author\" content=\"Pooja\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:creator\" content=\"@indcareer\" \/>\n<meta name=\"twitter:site\" content=\"@indcareer\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Pooja\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"20 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/\"},\"author\":{\"name\":\"Pooja\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/#\/schema\/person\/d6945cf059726f162259ba738092301e\"},\"headline\":\"KC Sinha: Exercise 8.4 – Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f\",\"datePublished\":\"2023-09-08T01:41:32+00:00\",\"dateModified\":\"2023-09-08T01:44:19+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/\"},\"wordCount\":5726,\"publisher\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/#organization\"},\"image\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg\",\"articleSection\":[\"Class 10\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/\",\"url\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/\",\"name\":\"KC Sinha: Exercise 8.4 - Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f - IndCareer Schools\",\"isPartOf\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage\"},\"image\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage\"},\"thumbnailUrl\":\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg\",\"datePublished\":\"2023-09-08T01:41:32+00:00\",\"dateModified\":\"2023-09-08T01:44:19+00:00\",\"description\":\"Type-I : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u090f\u0935\u0902 \u092c\u0940\u091c\u0940\u092f \u0938\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u0940\u0927\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 : Question 1 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0930\u0947\u0902 : (i) $\\\\sin ^{2} \\\\theta+\\\\cos ^{2}\",\"breadcrumb\":{\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage\",\"url\":\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg\",\"contentUrl\":\"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg\",\"width\":1600,\"height\":901,\"caption\":\"KC Sinha: Exercise 8.4 - 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Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f - IndCareer Schools","description":"Type-I : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u090f\u0935\u0902 \u092c\u0940\u091c\u0940\u092f \u0938\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u0940\u0927\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 : Question 1 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0930\u0947\u0902 : (i) $sin ^{2} theta+cos ^{2}","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","og_locale":"en_US","og_type":"article","og_title":"KC Sinha: Exercise 8.4 - Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f","og_description":"Type-I : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u090f\u0935\u0902 \u092c\u0940\u091c\u0940\u092f \u0938\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u0940\u0927\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 : Question 1 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0930\u0947\u0902 : (i) $sin ^{2} theta+cos ^{2}","og_url":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","og_site_name":"IndCareer Schools","article_publisher":"https:\/\/www.facebook.com\/indcareer","article_published_time":"2023-09-08T01:41:32+00:00","article_modified_time":"2023-09-08T01:44:19+00:00","og_image":[{"width":1600,"height":901,"url":"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg","type":"image\/jpeg"}],"author":"Pooja","twitter_card":"summary_large_image","twitter_creator":"@indcareer","twitter_site":"@indcareer","twitter_misc":{"Written by":"Pooja","Est. reading time":"20 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#article","isPartOf":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/"},"author":{"name":"Pooja","@id":"https:\/\/www.indcareer.com\/schools\/#\/schema\/person\/d6945cf059726f162259ba738092301e"},"headline":"KC Sinha: Exercise 8.4 – Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f","datePublished":"2023-09-08T01:41:32+00:00","dateModified":"2023-09-08T01:44:19+00:00","mainEntityOfPage":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/"},"wordCount":5726,"publisher":{"@id":"https:\/\/www.indcareer.com\/schools\/#organization"},"image":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage"},"thumbnailUrl":"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg","articleSection":["Class 10"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","url":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","name":"KC Sinha: Exercise 8.4 - Mathematics Solution Class 10 Chapter 8 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u090f\u0935\u092e \u0938\u0930\u094d\u0935\u0938\u092e\u093f\u0915\u093e\u090f - IndCareer Schools","isPartOf":{"@id":"https:\/\/www.indcareer.com\/schools\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage"},"image":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage"},"thumbnailUrl":"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg","datePublished":"2023-09-08T01:41:32+00:00","dateModified":"2023-09-08T01:44:19+00:00","description":"Type-I : \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u090f\u0935\u0902 \u092c\u0940\u091c\u0940\u092f \u0938\u0941\u0924\u094d\u0930\u094b\u0902 \u0915\u0947 \u0938\u0940\u0927\u0947 \u092a\u094d\u0930\u092f\u094b\u0917 \u092a\u0930 \u0906\u0927\u093e\u0930\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928 : Question 1 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0930\u0947\u0902 : (i) $\\sin ^{2} \\theta+\\cos ^{2}","breadcrumb":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-4-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage","url":"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg","contentUrl":"https:\/\/www.indcareer.com\/schools\/wp-content\/uploads\/2023\/09\/indcareer-schools-2-7-scaled.jpg","width":1600,"height":901,"caption":"KC Sinha: Exercise 8.4 - 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