{"id":625696,"date":"2023-09-08T01:17:34","date_gmt":"2023-09-08T01:17:34","guid":{"rendered":"https:\/\/www.indcareer.com\/schools\/?p=625696"},"modified":"2023-09-08T01:36:37","modified_gmt":"2023-09-08T01:36:37","slug":"kc-sinha-exercise-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye","status":"publish","type":"post","link":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","title":{"rendered":"KC Sinha: Exercise 8.3 – 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

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u094b 90\u00b0-\u03b8 \u0915\u0947 \u092a\u0942\u0930\u0915 \u0915\u094b\u0923 \u0915\u0940 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0940\u091c\u093f\u090f \u0964<\/strong><\/p>\n\n\n\n

(i) tan(90\u00b0-\u03b8)<\/strong><\/p>\n\n\n\n

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

cot \u03b8<\/p>\n\n\n\n

(ii) cos(90\u00b0-\u03b8)<\/strong><\/p>\n\n\n\n

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

sin \u03b8<\/p>\n\n\n\n

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

\u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0930\u094d\u0924\u093f 0\u00b0 \u0914\u0930 90\u00b0 \u0915\u0947 \u092c\u0940\u091a \u0915\u0947 \u0915\u093f\u0938\u0940 \u0915\u094b\u0923 \u0938\u0947 \u0915\u0930\u0947\u0902\u0964<\/strong><\/p>\n\n\n\n

(i) $\\sin 70^{\\circ}=\\cos (\\ldots)$<\/strong><\/p>\n\n\n\n

Sol :
cos(90\u00b0-70\u00b0)=20\u00b0<\/p>\n\n\n\n

(ii) $\\sin 35^{\\circ}=\\cos (\\ldots)$<\/strong><\/p>\n\n\n\n

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

$\\cos \\left(90^{\\circ}-35^{\\circ}\\right)=55^{\\circ}$<\/p>\n\n\n\n

(iii) $\\cos 48^{\\circ}=\\sin (\\ldots)$<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-48^{\\circ}\\right)=42^{\\circ}$<\/p>\n\n\n\n

(iv) $\\cos 70^{\\circ}=\\sin (\\ldots)$<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-70^{\\circ}\\right)=20^{\\circ}$<\/p>\n\n\n\n

(v) $\\cos 50^{\\circ}=\\sin (\\ldots)$<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-50^{\\circ}\\right)=40^{\\circ}$<\/p>\n\n\n\n

(vi) $\\sec 32^{\\circ}=\\operatorname{cosec}(\\ldots)$<\/strong><\/p>\n\n\n\n

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

$\\operatorname{cosec}\\left(90^{\\circ}-32^{\\circ}\\right)=58^{\\circ}$<\/p>\n\n\n\n

TYPE II: \u0915\u094b\u0923 \u03b8 \u0915\u0947 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u093e\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u0926\u093f\u090f \u0930\u0939\u0928\u0947 \u092a\u0930 \u0928\u094d\u092f\u0942\u0928\u0915\u094b\u0923 \u03b8 \u0915\u0947 \u092a\u0942\u090f\u0915 \u0915\u094b\u0923 \u0915\u0947 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924\u094b\u0902 \u0915\u094b \u091c\u094d\u091e\u093e\u0924 \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 4<\/h4>\n\n\n\n

\u092f\u0926\u093f $\\mathrm{A}+\\mathrm{B}=90^{\\circ}$, \u0924\u094b \u092a\u0942\u0930\u0915 \u0915\u094b\u0923 $\\mathrm{A}$ \u092f\u093e B \u0915\u0947 \u0909\u092a\u092f\u0941\u0915\u094d\u0924 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u093e\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u0938\u0947 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0942\u0930\u094d\u0924\u093f \u0915\u0940\u091c\u093f\u090f\u0964<\/strong><\/p>\n\n\n\n

(i) sin A=….<\/strong><\/p>\n\n\n\n

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

$=\\sin \\left(90^{\\circ}-B\\right)=\\cos B$<\/p>\n\n\n\n

(ii) cos B=…<\/strong><\/p>\n\n\n\n

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

$=\\cos \\left(90^{\\circ}-A\\right)=\\cos A$<\/p>\n\n\n\n

(iii) sec A=…<\/strong><\/p>\n\n\n\n

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

$=\\sec \\left(90^{\\circ}-\\mathrm{B}\\right)=\\operatorname{cosec} \\mathrm{B}$<\/p>\n\n\n\n

(iv) tan B=…<\/strong><\/p>\n\n\n\n

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

$=\\tan \\left(90^{\\circ}-\\mathrm{A}\\right)=\\cot \\mathrm{A}$<\/p>\n\n\n\n

(v) cosec B=…<\/strong><\/p>\n\n\n\n

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

$=\\operatorname{cosec}\\left(90^{\\circ}-\\mathrm{A}\\right)=\\sec \\mathrm{A}$<\/p>\n\n\n\n

(vi) cot A=…<\/strong><\/p>\n\n\n\n

$=\\cot \\left(90^{\\circ}-B\\right)=\\tan B$<\/p>\n\n\n\n

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

(i) \u092f\u0926\u093f $\\sin 37^{\\circ}=a$, \u0924\u092c $\\cos 53^{\\circ}$ \u0915\u093e \u092e\u093e\u0928 a \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

$\\sin 37^{\\circ}=a$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-37^{\\circ}\\right)=a$<\/p>\n\n\n\n

$\\cos 53^{\\circ}=\\mathrm{a}$<\/p>\n\n\n\n

(ii) \u092f\u0926\u093f $\\cos 47^{\\circ}=a$, \u0924\u092c $\\sin 43^{\\circ}$ \u0915\u093e \u092e\u093e\u0928 a \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 <\/strong>\u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930\u0947\u0902\u0964<\/strong><\/p>\n\n\n\n

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

$\\cos 47^{\\circ}=a$<\/p>\n\n\n\n

$\\sin \\left(90^{\\circ}-47^{\\circ}\\right)=a$<\/p>\n\n\n\n

$\\sin 43^{\\circ}=a$<\/p>\n\n\n\n

(iii) \u092f\u0926\u093f $\\sin 52^{\\circ}=a$, \u0924\u092c $\\cos 38^{\\circ}$ \u0915\u093e \u092e\u093e\u0928 a \u0915\u0947 \u092a\u0926\u094b\u0902 \u092e\u0947\u0902 <\/strong>\u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0930\u0947\u0902\u0964<\/strong><\/p>\n\n\n\n

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

$\\sin 52^{\\circ}=a$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-52^{\\circ}\\right)=a$<\/p>\n\n\n\n

$\\cos 38^{\\circ}=a$<\/p>\n\n\n\n

(iv) \u092f\u0926\u093f $\\sin 56^{\\circ}=x$, \u0924\u092c $\\sin 34^{\\circ}$ \u0915\u093e \u092e\u093e\u0928 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

$\\sin 56^{\\circ}=x$ <\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-56^{\\circ}\\right)=x$ <\/p>\n\n\n\n

$\\cos 34^{\\circ}=x$<\/p>\n\n\n\n

Type III: \u092a\u0942\u0930\u0915 \u0915\u094b\u0923\u094b\u0902 \u0915\u0947 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092d\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924\u094b\u0902 \u0938\u0947 \u0938\u092e\u094d\u092c\u0930\u094d\u0926\u094d\u0927 \u0935\u094d\u092f\u0902\u091c\u0915\u094b\u0902 \u0915\u0947 \u092e\u093e\u0928\u094b\u0902 \u092a\u0930 \u0906\u092f\u093e\u090f\u093f\u0924 \u092a\u094d\u0930\u0936\u094d\u0928:<\/strong><\/p>\n\n\n\n

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

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u0947 \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0930\u0947\u0902\u0964<\/strong><\/p>\n\n\n\n

(i) $\\frac{\\cos 59^{\\circ}}{\\sin 31^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin \\left(90^{\\circ}-59\\right)}{\\sin 31^{\\circ}}=\\frac{\\sin 31^{\\circ}}{\\sin 31^{\\circ}}=1$<\/p>\n\n\n\n

(ii) $\\frac{\\cos 53^{\\circ}}{\\sin 37^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin \\left(90^{\\circ}-53^{\\circ}\\right)}{\\sin 37^{\\circ}}=\\frac{\\sin 37^{\\circ}}{\\sin 37^{\\circ}}=1$<\/p>\n\n\n\n

(iii) $\\frac{\\sin 20^{\\circ}}{\\cos 70^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cos \\left(90^{\\circ}-20^{\\circ}\\right)}{\\cos 70^{\\circ}}=\\frac{\\cos 70^{\\circ}}{\\cos 70^{\\circ}}=1$<\/p>\n\n\n\n

(iv) $\\frac{\\sqrt{2} \\sin 22^{\\circ}}{\\cos 68^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sqrt{2} \\cos \\left(90^{\\circ}-22^{\\circ}\\right)}{\\cos 68^{\\circ}}=\\frac{\\sqrt{2} \\cos 68^{\\circ}}{\\operatorname{cos} 68^{\\circ}}=\\sqrt{2} \\times 1=\\sqrt{2}$<\/p>\n\n\n\n

(v) $\\frac{\\sin 10^{\\circ}}{\\cos 80^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cos \\left(90^{\\circ}-10^{\\circ}\\right)}{\\cos 80^{\\circ}}=\\frac{\\cos 80^{\\circ}}{\\cos 80^{\\circ}}=1$<\/p>\n\n\n\n

(vi) $\\frac{\\sin 27^{\\circ}}{\\cos 63^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cos \\left(90^{\\circ}-27^{\\circ}\\right)}{\\cos 63^{\\circ}}=\\frac{\\cos 63^{\\circ}}{\\cos 63^{\\circ}}=1$<\/p>\n\n\n\n

(vii) $\\frac{\\sqrt{3} \\cos 65^{\\circ}}{\\sin 25^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sqrt{3} \\sin \\left(90^{\\circ}-65^{\\circ}\\right)}{\\sin 25^{\\circ}}$<\/p>\n\n\n\n

$=\\frac{\\sqrt{3} \\sin 25^{\\circ}}{\\sin 25^{\\circ}}=\\sqrt{3} \\times 1=\\sqrt{3}$<\/p>\n\n\n\n

(viii) $\\frac{\\cos 29^{\\circ}}{\\sin 61^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin \\left(90^{\\circ}-29^{\\circ}\\right)}{\\sin 61^{\\circ}}=\\frac{\\sin 61^{\\circ}}{\\sin 61^{\\circ}}=1$<\/p>\n\n\n\n

(ix) $\\sin 54^{\\circ}-\\cos 36^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\cos \\left(90^{\\circ}-54^{\\circ}\\right)-\\cos 36^{\\circ}$<\/p>\n\n\n\n

$\\cos 36^{\\circ}-\\cos 36^{\\circ}=0$<\/p>\n\n\n\n

(x) $\\frac{\\tan 80^{\\circ}}{\\cot 10^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cot \\left(90^{\\circ}-80^{\\circ}\\right)}{\\cot 10^{\\circ}}=\\frac{\\cot 10^{\\circ}}{\\operatorname{cot} 10^{\\circ}}=1$<\/p>\n\n\n\n

(xi) $ \\operatorname{cosec} 31^{\\circ}-\\sec 59^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\sec \\left(90^{\\circ}-31^{\\circ}\\right)-\\sec 59^{\\circ}$<\/p>\n\n\n\n

$\\sec 59^{\\circ}-\\sec 59^{\\circ}=0$<\/p>\n\n\n\n

(xii) $\\frac{\\sin 18^{\\circ}}{\\cos 72^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cos \\left(90^{\\circ}-18^{\\circ}\\right)}{\\cos 72^{\\circ}}=\\frac{\\cos 72^{\\circ}}{\\cos 72^{\\circ}}=1$<\/p>\n\n\n\n

(xiii) $\\frac{\\tan 65^{\\circ}}{\\cos 25^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cot \\left(90^{\\circ}-65^{\\circ}\\right)}{\\cot 25^{\\circ}}=\\frac{\\cot 25^{\\circ}}{\\cot 25^{\\circ}}=1$<\/p>\n\n\n\n

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

\u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u094b \u092d\u0930\u0947\u0902-<\/strong><\/p>\n\n\n\n

(i) \u092f\u0926\u093f $\\sin 50^{\\circ}=0.7660$, \u0924\u094b $\\cos 40^{\\circ}=\\ldots \\ldots . .$<\/strong><\/p>\n\n\n\n

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

$\\cos \\left(90^{\\circ}-50^{\\circ}\\right)=0.7660$<\/p>\n\n\n\n

$\\cos 40^{\\circ}=0.7660$<\/p>\n\n\n\n

(ii) \u092f\u0926\u093f $\\cos 44^{\\circ}=0.7193$, \u0924\u094b $\\sin 46^{\\circ}=$.<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-44^{\\circ}\\right)=0.7193$<\/p>\n\n\n\n

$\\sin 46^{\\circ}=0.7193$<\/p>\n\n\n\n

(iii) $\\sin 50^{\\circ}+\\cos 40^{\\circ}=2 \\sin (\\ldots \\ldots . .)$<\/strong><\/p>\n\n\n\n

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

$\\sin 50^{\\circ}+\\cos 40^{\\circ}$<\/p>\n\n\n\n

$\\sin 50^{\\circ}+\\sin \\left(90^{\\circ}+40^{\\circ}\\right)$<\/p>\n\n\n\n

$\\sin 50^{\\circ}+\\sin 50^{\\circ}$<\/p>\n\n\n\n

$2 \\sin 50^{\\circ}$<\/p>\n\n\n\n

(iv) $\\frac{\\sin 70^{\\circ}}{\\cos 20^{\\circ}}$ \u0915\u093e \u092e\u093e\u0928 $\\ldots \\ldots$ \u0939\u0948<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cos \\left(90^{\\circ}-70^{\\circ}\\right)}{\\cos 20^{\\circ}}=\\frac{\\cos 20^{\\circ}}{\\cos 20^{\\circ}}=1$<\/p>\n\n\n\n

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

(i) \u092f\u0926\u093f $\\mathrm{A}+\\mathrm{B}=90^{\\circ}$, \u0924\u092c $\\cos \\mathrm{B}$ \u0915\u094b $\\mathrm{A}$ \u0915\u0947 \u0938\u0930\u0932\u0924\u092e \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \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

$\\cos B=\\cos \\left(90^{\\circ}-A\\right)=\\sin A$<\/p>\n\n\n\n

(ii) \u092f\u0926\u093f $\\mathrm{X}+\\mathrm{Y}=90^{\\circ}$, \u0924\u092c $\\cos \\mathrm{X}$ \u0915\u094b $\\mathrm{Y}$ \u0915\u0947 \u0938\u0930\u0932\u0924\u092e \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \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

$\\cos X=\\cos \\left(90^{\\circ}-Y\\right)=\\sin Y$<\/p>\n\n\n\n

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

(i) \u092f\u0926\u093f $\\mathrm{A}+\\mathrm{B}=90^{\\circ}, \\sin \\mathrm{A}=a, \\sin \\mathrm{B}=b$, \u0924\u094b \u0938\u093f\u0926\u094d\u0927 \u0915\u0940\u091c\u093f\u090f \u0915\u093f-<\/strong><\/p>\n\n\n\n

(a) $a^{2}+b^{2}=1$<\/strong><\/p>\n\n\n\n

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

sin A=a, sin B=b<\/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

$\\begin{aligned}&\\sin ^{2} \\mathrm{~A}+\\sin ^{2} \\mathrm{~B}=a^{2}+b^{2} \\\\&\\cos ^{2}\\left(90^{\\circ}-\\mathrm{A}\\right)+\\cos ^{2}\\left(90^{\\circ}-\\mathrm{B}\\right)=a^{2}+b^{2} \\\\&\\cos ^{2} \\mathrm{~B}+\\sin ^{2} \\mathrm{~A}=a^{2}+b^{2} \\\\&1=a^{2}+b^{2}\\end{aligned}$<\/p>\n\n\n\n

(b) $\\tan \\mathrm{A}=\\frac{a}{b}$<\/strong><\/p>\n\n\n\n

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

sin A=a, sin B=b<\/p>\n\n\n\n

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

$\\begin{aligned}&\\frac{\\sin \\mathrm{A}}{\\sin \\mathrm{B}}=\\frac{a}{b} \\\\&\\frac{\\sin \\mathrm{A}}{\\cos \\left(90^{\\circ}-\\mathrm{B}\\right)}=\\frac{a}{b} \\\\&\\frac{\\sin \\mathrm{A}}{\\cos \\mathrm{A}}=\\frac{a}{b} \\\\&\\tan =\\frac{a}{b}\\end{aligned}$<\/p>\n\n\n\n

(ii) \u0926\u093f\u0916\u093e\u092f\u0947\u0902 \u0915\u093f $\\sin \\left(50^{\\circ}+\\theta\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)=0$.<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(50^{\\circ}+\\theta\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)=0$<\/p>\n\n\n\n

$\\cos \\left[90^{\\circ}-\\left(50^{\\circ}+\\theta\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)\\right.$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-50^{\\circ}-\\theta\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

$\\cos \\left(40^{\\circ}-\\theta\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)$=0<\/p>\n\n\n\n

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

\u0938\u093f\u0926\u094d\u0927 \u0915\u0940\u091c\u093f\u090f \u0915\u093f $\\frac{\\cos \\theta}{\\sin \\left(90^{\\circ}-\\theta\\right)}+\\frac{\\sin \\theta}{\\cos \\left(90^{\\circ}-\\theta\\right)}=2$<\/strong><\/p>\n\n\n\n

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

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

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

1+1=2<\/p>\n\n\n\n

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

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

\u0915\u093f\u0938\u0940 $\\triangle \\mathrm{ABC}$ \u092e\u0947\u0902 \u0938\u093f\u0926\u094d\u0927 \u0915\u0940\u091c\u093f\u090f \u0915\u093f-<\/strong><\/p>\n\n\n\n

(a) $\\sin \\frac{\\mathrm{B}+\\mathrm{C}}{2}=\\cos \\frac{\\mathrm{A}}{2}$<\/strong><\/p>\n\n\n\n

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

\u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u0947 \u0924\u0940\u0928\u094b\u0902 \u0915\u094b\u0923\u094b \u0915\u093e \u092e\u093e\u0928 $180^{\\circ}$ \u0939\u094b\u0924\u093e \u0939\u0948<\/p>\n\n\n\n

$A+B+C=180^{\\circ}$<\/p>\n\n\n\n

$B+C=180^{\\circ}-A$<\/p>\n\n\n\n

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

$\\sin \\frac{B+C}{2}=\\cos \\frac{A}{2}$<\/p>\n\n\n\n

$\\sin \\left(\\frac{180^{\\circ}-\\mathrm{A}}{2}\\right)=\\cos \\frac{\\mathrm{A}}{2}$<\/p>\n\n\n\n

$\\sin \\left(\\frac{180^{\\circ}}{2}-\\frac{A}{2}\\right)=\\cos \\frac{A}{2}$<\/p>\n\n\n\n

$\\sin \\left(90^{\\circ}-\\frac{A}{2}\\right)=\\cos \\frac{A}{2}$<\/p>\n\n\n\n

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

(b) $\\tan \\frac{\\mathrm{B}+\\mathrm{C}}{2}=\\cot \\frac{\\mathrm{A}}{2}$<\/strong><\/p>\n\n\n\n

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

\u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u0947 \u0924\u0940\u0928\u094b\u0902 \u0915\u094b\u0923\u094b \u0915\u093e \u092e\u093e\u0928 $180^{\\circ}$ \u0939\u094b\u0924\u093e \u0939\u0948<\/p>\n\n\n\n

$\\begin{aligned}&A+B+C=180^{\\circ} \\\\&B+C=180^{\\circ}-A\\end{aligned}$<\/p>\n\n\n\n

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

$\\tan \\frac{B+C}{2}=\\cot \\frac{A}{2}$<\/p>\n\n\n\n

$\\tan \\left(\\frac{180^{\\circ}-A}{2}\\right)=\\cot \\frac{A}{2}$<\/p>\n\n\n\n

$\\tan \\left(\\frac{180^{\\circ}}{2}-\\frac{A}{2}\\right)=\\cot \\frac{A}{2}$<\/p>\n\n\n\n

$\\tan \\left(90^{\\circ}-\\frac{A}{2}\\right)=\\cot \\frac{A}{2}$<\/p>\n\n\n\n

$\\cot \\frac{A}{2}=\\cot \\frac{A}{2}$<\/p>\n\n\n\n

(c) $\\cos \\frac{\\mathrm{A}+\\mathrm{B}}{2}=\\sin \\frac{\\mathrm{C}}{2}$<\/strong><\/p>\n\n\n\n

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

\u0924\u094d\u0930\u093f\u092d\u0941\u091c \u0915\u0947 \u0924\u0940\u0928\u094b\u0902 \u0915\u094b\u0923\u094b \u0915\u093e \u092e\u093e\u0928 $180^{\\circ}$ \u0939\u094b\u0924\u093e \u0939\u0948<\/p>\n\n\n\n

$\\begin{aligned}&A+B+C=180^{\\circ} \\\\&A+B=180^{\\circ}-A\\end{aligned}$<\/p>\n\n\n\n

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

$\\begin{aligned}&\\cos \\frac{A+B}{2}=\\sin \\frac{A}{2} \\\\&\\cos \\left(\\frac{180^{\\circ}-C}{2}\\right)=\\sin \\frac{C}{2}\\end{aligned}$<\/p>\n\n\n\n

$\\cos \\frac{A+B}{2}=\\sin \\frac{A}{2}$<\/p>\n\n\n\n

$\\cos \\left(\\frac{180^{\\circ}-C}{2}\\right)=\\sin \\frac{C}{2}$<\/p>\n\n\n\n

$\\cos \\left(\\frac{180^{\\circ}}{2}-\\frac{C}{2}\\right)=\\sin \\frac{C}{2}$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-\\frac{\\mathrm{C}}{2}\\right)=\\sin \\frac{\\mathrm{C}}{2}$<\/p>\n\n\n\n

$\\sin \\frac{\\mathrm{C}}{2}=\\sin \\frac{\\mathrm{C}}{2}$<\/p>\n\n\n\n

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

(i) \u092f\u0926\u093f $\\sin 3 \\mathrm{~A}=\\cos \\left(\\mathrm{A}-26^{\\circ}\\right)$, \u091c\u0939\u093e\u0901 $3 \\mathrm{~A}$ \u090f\u0915 \u0928\u094d\u092f\u0942\u0928\u0915\u094b\u0923 \u0939\u0948 \u0924\u092c $\\mathrm{A}$ \u0915\u093e \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f \u0964<\/strong><\/p>\n\n\n\n

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

$\\cos \\left(90^{\\circ}-3 A\\right)=\\cos \\left(A-26^{\\circ}\\right)$<\/p>\n\n\n\n

$90^{\\circ}-3 A=A-26^{\\circ}$<\/p>\n\n\n\n

$90^{\\circ}+26^{\\circ}=A+3 A$<\/p>\n\n\n\n

$116^{\\circ}=4 A$<\/p>\n\n\n\n

$A=\\frac{116}{4}$<\/p>\n\n\n\n

$A=29^{\\circ}$<\/p>\n\n\n\n

(ii) \u092f\u0926\u093f $\\cos \\left(2 \\theta+54^{\\circ}\\right)=\\sin \\theta$. \u091c\u0939\u093e\u0901 $\\left(2 \\theta+54^{\\circ}\\right)$ \u090f\u0915 \u0928\u094d\u092f\u0942\u0928\u0915\u094b\u0923 \u0939\u0948 \u0924\u092c $\\theta$ \u0915\u093e \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f \u0964<\/strong><\/p>\n\n\n\n

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

$\\cos \\left(2 \\theta+54^{\\circ}\\right)=\\cos \\left(90^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

$2 \\theta+54^{\\circ}=90^{\\circ}-\\theta$<\/p>\n\n\n\n

$2 \\theta+\\theta=90^{\\circ}-54^{\\circ}$<\/p>\n\n\n\n

$3 \\theta=36^{\\circ}$<\/p>\n\n\n\n

$\\theta=\\frac{36^{\\circ}}{3}$<\/p>\n\n\n\n

$\\theta=12^{\\circ}$<\/p>\n\n\n\n

(iii) \u092f\u0926\u093f $\\tan 3 \\theta=\\cot \\left(\\theta+18^{\\circ}\\right)$, \u091c\u0939\u093e\u0901 $3 \\theta$ \u0914\u0930 $\\theta+18^{\\circ}$ \u0928\u094d\u092f\u0942\u0928\u0915\u094b\u0923 \u0939\u0948\u0902, \u0924\u094b $\\theta$ \u0915\u093e \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f \u0964<\/strong><\/p>\n\n\n\n

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

$\\cot =\\left(90^{\\circ}-3 \\theta\\right)=\\cot \\left(\\theta+18^{\\circ}\\right)$<\/p>\n\n\n\n

$90^{\\circ}-3 \\theta=\\theta+18^{\\circ}$<\/p>\n\n\n\n

$-3 \\theta-\\theta=18^{\\circ}-90^{\\circ}$<\/p>\n\n\n\n

$-4 \\theta=-72^{\\circ}$<\/p>\n\n\n\n

$\\theta=\\frac{72}{4}$<\/p>\n\n\n\n

$\\theta=18^{\\circ}$<\/p>\n\n\n\n

(iv) \u092f\u0926\u093f $\\sec 5 \\theta=\\operatorname{cosec}\\left(\\theta-36^{\\circ}\\right)$, \u091c\u0939\u093e\u0901 5\u03b8 \u090f\u0915 \u0928\u094d\u092f\u0942\u0928\u0915\u094b\u0923 \u0939\u0948, \u0924\u094b \u03b8 \u0915\u093e \u092e\u093e\u0928 \u0928\u093f\u0915\u093e\u0932\u093f\u090f \u0964<\/strong><\/p>\n\n\n\n

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

$\\operatorname{cosec}\\left(90^{\\circ}-5 \\theta\\right)=\\operatorname{cosec}\\left(\\theta-36^{\\circ}\\right)$<\/p>\n\n\n\n

$90^{\\circ}-5 \\theta=\\theta-36^{\\circ}$<\/p>\n\n\n\n

$90^{\\circ}+36^{\\circ}=\\theta+5 \\theta$<\/p>\n\n\n\n

$126^{\\circ}=6 \\theta$<\/p>\n\n\n\n

$6 \\theta=126^{\\circ}$<\/p>\n\n\n\n

$\\theta=\\frac{126}{6}$<\/p>\n\n\n\n

$\\theta=21^{\\circ}$<\/p>\n\n\n\n

\u0938\u093f\u0926\u094d\u0927 \u0915\u0940\u091c\u093f\u090f \u0915\u093f :<\/strong><\/p>\n\n\n\n

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

$\\sin 70^{\\circ} \\cdot \\sec 20^{\\circ}=1$<\/strong><\/p>\n\n\n\n

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

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

$\\sin 70^{\\circ} . \\operatorname{Sec} 20^{\\circ}$<\/p>\n\n\n\n

$\\sin 70^{\\circ} \\cdot \\operatorname{cosec}\\left(90^{\\circ}-20^{\\circ}\\right)$<\/p>\n\n\n\n

$\\sin 70^{\\circ} \\cdot \\operatorname{cosec} 70^{\\circ}=1$ R.H.S<\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-\\theta\\right) \\tan \\theta=\\sin \\theta$<\/strong><\/p>\n\n\n\n

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

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

$\\sin \\left(90^{\\circ}-\\theta\\right) \\cdot \\tan \\theta$<\/p>\n\n\n\n

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

$=\\sin \\theta$ R.H.S<\/p>\n\n\n\n

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

$\\tan 63^{\\circ} \\cdot \\tan 27^{\\circ}=1$<\/strong><\/p>\n\n\n\n

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

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

$\\tan 63^{\\circ} . \\tan 27$<\/p>\n\n\n\n

$\\tan 63^{\\circ} \\cdot \\cot \\left(90^{\\circ}-27^{\\circ}\\right)$<\/p>\n\n\n\n

$\\tan 63^{\\circ} \\cdot \\cot 63^{\\circ}$<\/p>\n\n\n\n

=1 R.H.S<\/p>\n\n\n\n

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

$\\frac{\\sin \\left(90^{\\circ}-\\theta\\right) \\sin \\theta}{\\tan \\theta}-1=-\\sin ^{2} \\theta$<\/strong><\/p>\n\n\n\n

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

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

$<\/p>\n\n\n\n

\\begin{aligned}<\/p>\n\n\n\n

&\\frac{\\sin \\left(90^{\\circ}-\\theta\\right) \\sin \\theta}{\\tan \\theta}-1 \\\\<\/p>\n\n\n\n

&\\frac{\\cos \\theta \\cdot \\sin \\theta}{\\frac{\\sin \\theta}{\\cos \\theta}}-1 \\\\<\/p>\n\n\n\n

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

&-\\sin ^{2} \\theta \\text { R.H.S }<\/p>\n\n\n\n

\\end{aligned}<\/p>\n\n\n\n

$<\/p>\n\n\n\n

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

$\\sin 55^{\\circ} \\cdot \\cos 48^{\\circ}=\\cos 35^{\\circ} \\cdot \\sin 42^{\\circ}$<\/strong><\/p>\n\n\n\n

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

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

$\\sin 55^{\\circ} \\cdot \\cos 48^{\\circ}$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-55^{\\circ}\\right) \\cdot \\sin \\left(90^{\\circ}-48^{\\circ}\\right)$<\/p>\n\n\n\n

$\\cos 35^{\\circ} . \\operatorname{Sin} 42^{\\circ} \\quad$ R.H.S<\/p>\n\n\n\n

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

$\\sin 25^{\\circ} \\cdot \\sin 65^{\\circ}=\\cos 25^{\\circ} \\cdot \\cos 65^{\\circ}$<\/strong><\/p>\n\n\n\n

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

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

$\\sin 25^{\\circ} . \\operatorname{Sin} 65^{\\circ}$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-25^{\\circ}\\right) \\cdot \\cos \\left(90^{\\circ}-65^{\\circ}\\right)$<\/p>\n\n\n\n

$\\cos 25^{\\circ} \\cdot \\cos 65^{\\circ} \\quad$ R.H.S<\/p>\n\n\n\n

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

$\\sin 54^{\\circ}+\\cos 67^{\\circ}=\\sin 23^{\\circ}+\\cos 36^{\\circ}$<\/strong><\/p>\n\n\n\n

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

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

$\\sin 54^{\\circ} \\cdot \\cos 67^{\\circ}$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-54^{\\circ}\\right) \\cdot \\sin \\left(90^{\\circ}-67^{\\circ}\\right)$<\/p>\n\n\n\n

$\\sin 23^{\\circ} \\cdot \\cos 36^{\\circ} \\quad$ R.H.S<\/p>\n\n\n\n

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

$\\cos 27^{\\circ}+\\sin 51^{\\circ}=\\sin 63^{\\circ}+\\cos 39^{\\circ}$<\/strong><\/p>\n\n\n\n

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

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

$\\cos 27^{\\circ} \\cdot \\sin 51^{\\circ}$<\/p>\n\n\n\n

$\\sin \\left(90^{\\circ}-27^{\\circ}\\right) \\cdot \\cos \\left(90^{\\circ}-51^{\\circ}\\right)$<\/p>\n\n\n\n

$\\sin 63^{\\circ} \\cdot \\cos 39^{\\circ} \\quad$ R.H.S<\/p>\n\n\n\n

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

$\\sin ^{2} 40^{\\circ}+\\sin ^{2} 50^{\\circ}=1$<\/strong><\/p>\n\n\n\n

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

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

$<\/p>\n\n\n\n

\\begin{aligned}<\/p>\n\n\n\n

&\\sin ^{2} 40^{\\circ}+\\sin ^{2} 50^{\\circ} \\\\<\/p>\n\n\n\n

&\\sin ^{2} 40^{\\circ}+\\cos ^{2}\\left(90^{\\circ}-50^{\\circ}\\right) \\\\<\/p>\n\n\n\n

&\\sin ^{2} 40^{\\circ}+\\cos ^{2} 40^{\\circ} \\\\<\/p>\n\n\n\n

&=1 \\quad \\text { R.H.S }<\/p>\n\n\n\n

\\end{aligned}<\/p>\n\n\n\n

$<\/p>\n\n\n\n

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

$\\sin ^{2} 29^{\\circ}+\\sin ^{2} 61^{\\circ}=1$<\/strong><\/p>\n\n\n\n

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

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

$<\/p>\n\n\n\n

\\begin{aligned}<\/p>\n\n\n\n

&\\sin ^{2} 29^{\\circ}+\\sin ^{2} 61^{\\circ} \\\\<\/p>\n\n\n\n

&\\sin ^{2} 29^{\\circ}+\\cos ^{2}\\left(90^{\\circ}-61^{\\circ}\\right)<\/p>\n\n\n\n

\\end{aligned}<\/p>\n\n\n\n

$<\/p>\n\n\n\n

$\\sin ^{2} 29^{\\circ}+\\cos ^{2} 29^{\\circ}$<\/p>\n\n\n\n

=1 R.H.S<\/p>\n\n\n\n

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

$\\sin \\theta \\cdot \\cos \\left(90^{\\circ}-\\theta\\right)+\\cos \\theta \\sin \\left(90^{\\circ}-\\theta\\right)=1$<\/strong><\/p>\n\n\n\n

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

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

$<\/p>\n\n\n\n

\\begin{aligned}<\/p>\n\n\n\n

&\\sin \\theta \\cdot \\cos \\left(90^{\\circ}-\\theta\\right)+\\cos \\theta \\cdot \\sin \\left(90^{\\circ}-\\theta\\right) \\\\<\/p>\n\n\n\n

&\\sin \\theta \\times \\sin \\theta+\\cos \\theta \\times \\cos \\theta \\\\<\/p>\n\n\n\n

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

&=1 \\quad \\text { R.H.S }<\/p>\n\n\n\n

\\end{aligned}<\/p>\n\n\n\n

$<\/p>\n\n\n\n

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

$\\cos \\theta \\cdot \\cos \\left(90^{\\circ}-\\theta\\right)-\\sin \\theta \\cdot \\sin \\left(90^{\\circ}-\\theta\\right)=0$<\/strong><\/p>\n\n\n\n

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

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

$<\/p>\n\n\n\n

\\begin{aligned}<\/p>\n\n\n\n

&\\cos \\theta \\cdot \\cos \\left(90^{\\circ}-\\theta\\right)-\\sin \\theta \\cdot \\sin \\left(90^{\\circ}-\\theta\\right) \\\\<\/p>\n\n\n\n

&\\cos \\theta \\cdot \\sin \\theta-\\sin \\theta \\cdot \\cos \\theta \\\\<\/p>\n\n\n\n

&=0 \\quad \\text { R.H.S }<\/p>\n\n\n\n

\\end{aligned}<\/p>\n\n\n\n

$<\/p>\n\n\n\n

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

$\\sin 42^{\\circ} \\cdot \\cos 48^{\\circ}+\\cos 42^{\\circ} \\cdot \\sin 48^{\\circ}=1$<\/strong><\/p>\n\n\n\n

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

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

$\\sin 42^{\\circ} \\cdot \\cos 48^{\\circ}+\\cos 42^{\\circ} \\cdot \\sin 48^{\\circ}$<\/p>\n\n\n\n

$\\sin 42^{\\circ} \\cdot \\sin \\left(90^{\\circ}-48^{\\circ}\\right)+\\cos 42^{\\circ} \\cdot \\cos \\left(90^{\\circ}-48^{\\circ}\\right)$<\/p>\n\n\n\n

$\\sin 42^{\\circ} \\cdot \\sin 42^{\\circ}+\\cos 42^{\\circ} \\cdot \\cos 42^{\\circ}$<\/p>\n\n\n\n

=1 R.H.S<\/p>\n\n\n\n

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

$\\frac{\\cos 20^{\\circ}}{\\sin 70^{\\circ}}+\\frac{\\cos \\theta}{\\sin \\left(90^{\\circ}-\\theta\\right)}=2$<\/strong><\/p>\n\n\n\n

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

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

$<\/p>\n\n\n\n

\\begin{aligned}<\/p>\n\n\n\n

&\\frac{\\cos 20^{\\circ}}{\\sin 70^{\\circ}}+\\frac{\\cos \\theta}{\\sin \\left(90^{\\circ}-\\theta\\right)} \\\\<\/p>\n\n\n\n

&\\frac{\\cos 20^{\\circ}}{\\sin 70^{\\circ}}+\\frac{\\cos \\theta}{\\cos \\theta)} \\\\<\/p>\n\n\n\n

&\\frac{\\sin \\left(90^{\\circ}-20^{\\circ}\\right)}{\\sin 70^{\\circ}}+1 \\\\<\/p>\n\n\n\n

&\\frac{\\sin 70^{\\circ}}{\\sin 70^{\\circ}}+1 \\end{aligned}$<\/p>\n\n\n\n

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

$\\sin ^{2} 85^{\\circ}+\\sin ^{2} 5^{\\circ}+\\sin ^{2} 6 ?^{\\circ}+\\sin ^{2}, 23^{\\circ}=2$<\/strong><\/p>\n\n\n\n

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

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

$<\/p>\n\n\n\n

\\begin{aligned}<\/p>\n\n\n\n

&\\sin ^{2} 85^{\\circ}+\\sin ^{2} 5^{\\circ}+\\sin ^{2} 67^{\\circ}+\\sin ^{2} 23^{\\circ} \\\\<\/p>\n\n\n\n

&\\sin ^{2} 85^{\\circ}+\\cos ^{2}\\left(90^{\\circ}-5^{\\circ}\\right)+\\sin ^{2} 67^{\\circ}+\\cos ^{2}\\left(90^{\\circ}-23^{\\circ}\\right) \\\\<\/p>\n\n\n\n

&\\sin ^{2} 85^{\\circ}+\\cos ^{2} 85^{\\circ}+\\sin ^{2} 67^{\\circ}+\\cos ^{2} 67^{\\circ}<\/p>\n\n\n\n

\\end{aligned}<\/p>\n\n\n\n

$<\/p>\n\n\n\n

1+1=2 RHS<\/p>\n\n\n\n

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

$\\tan 9^{\\circ} \\cdot \\tan 27^{\\circ} \\cdot \\tan 45^{\\circ} \\cdot \\tan 63^{\\circ} \\cdot \\tan 81^{\\circ}=1$<\/strong><\/p>\n\n\n\n

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

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

$\\tan 9^{\\circ} \\cdot \\tan 27^{\\circ} \\cdot \\tan 45^{\\circ} \\cdot \\tan 63^{\\circ} \\cdot \\tan 81^{\\circ}$<\/p>\n\n\n\n

$\\left(\\tan 9^{\\circ} \\cdot \\tan 81^{\\circ}\\right) \\cdot\\left(\\tan 27^{\\circ} \\cdot \\tan 63^{\\circ}\\right) \\cdot \\tan 45^{\\circ}$<\/p>\n\n\n\n

$\\left(\\tan 9^{\\circ} \\cdot \\cot \\left(90^{\\circ}-81^{\\circ}\\right)\\right) \\cdot\\left(\\tan 27^{\\circ} \\cdot \\cot \\left(90^{\\circ}-63^{\\circ}\\right)\\right) \\cdot \\times 1$<\/p>\n\n\n\n

$\\left(\\tan 9^{\\circ} \\cdot \\cot 9^{\\circ}\\right) \\cdot\\left(\\tan 27^{\\circ} \\cdot \\cot 27^{\\circ}\\right) \\cdot \\times 1$<\/p>\n\n\n\n

=1\u00d71=1 RHS<\/p>\n\n\n\n

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

$\\sin 9^{\\circ} \\cdot \\sin 27^{\\circ} \\cdot \\sin 63^{\\circ} \\cdot \\sin ^{\\circ} 81^{\\circ}=\\cos 9^{\\circ} \\cdot \\cos 27^{\\circ} \\cdot \\cos 63^{\\circ} \\cdot \\cos 81^{\\circ}$<\/strong><\/p>\n\n\n\n

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

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

$\\sin 9^{\\circ} \\cdot \\sin 27^{\\circ} \\cdot \\sin 63^{\\circ} \\cdot \\sin 81^{\\circ}$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-9^{\\circ}\\right) \\cdot \\cos \\left(90^{\\circ}-27^{\\circ}\\right) \\cdot \\cos \\left(90^{\\circ}-63^{\\circ}\\right) \\cdot \\cos \\left(90^{\\circ}-81^{\\circ}\\right)$<\/p>\n\n\n\n

$\\cos 81^{\\circ} \\cdot \\cos 63^{\\circ} \\cdot \\cos 27^{\\circ} \\cdot \\cos 9^{\\circ}$<\/p>\n\n\n\n

$\\cos 9^{\\circ} \\cdot \\cos 27^{\\circ} \\cdot \\cos 63^{\\circ} \\cdot \\cos 81^{\\circ}$<\/p>\n\n\n\n

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

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

(i) $\\tan 7^{\\circ} \\cdot \\tan 23^{\\circ}, \\tan 60^{\\circ} \\cdot \\tan 67^{\\circ} \\cdot \\tan 83^{\\circ}=\\sqrt{3}$<\/strong><\/p>\n\n\n\n

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

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

$\\tan 7^{\\circ} \\cdot \\tan 23^{\\circ} \\cdot \\tan 60^{\\circ} \\cdot \\tan 67^{\\circ} \\cdot \\tan 83^{\\circ}$<\/p>\n\n\n\n

$\\left(\\tan 7^{\\circ} \\cdot \\tan 83^{\\circ}\\right) \\cdot\\left(\\tan 23^{\\circ} \\cdot \\tan 67^{\\circ}\\right) \\cdot \\tan 60^{\\circ}$<\/p>\n\n\n\n

$\\left(\\tan 7^{\\circ} \\cdot \\cot \\left(90^{\\circ}-83^{\\circ}\\right)\\right) \\cdot\\left(\\tan 23^{\\circ} \\cdot \\cot \\left(90^{\\circ}-67^{\\circ}\\right)\\right) \\cdot \\times \\sqrt{3}$<\/p>\n\n\n\n

$\\left(\\tan 7^{\\circ} \\cdot \\cot 7^{\\circ}\\right) \\cdot\\left(\\tan 23^{\\circ} \\cdot \\cot 23^{\\circ}\\right) \\cdot \\times \\sqrt{3}$<\/p>\n\n\n\n

$=1 \\times 1 \\times \\sqrt{3}=\\sqrt{3}$ R.H.S<\/p>\n\n\n\n

(ii) $\\tan 15^{\\circ} \\tan 25^{\\circ} \\tan 60^{\\circ} \\tan 65^{\\circ} \\tan 75^{\\circ}=\\sqrt{3}$<\/strong><\/p>\n\n\n\n

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

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

$\\tan 15^{\\circ} \\cdot \\tan 25^{\\circ} \\cdot \\tan 60^{\\circ} \\cdot \\tan 65^{\\circ} \\cdot \\tan 75^{\\circ}$<\/p>\n\n\n\n

$\\left(\\tan 15^{\\circ} \\cdot \\tan 75^{\\circ}\\right) \\cdot\\left(\\tan 25^{\\circ} \\cdot \\tan 65^{\\circ}\\right) \\cdot \\tan 60^{\\circ}$<\/p>\n\n\n\n

$\\left(\\tan 15^{\\circ} \\cdot \\cot \\left(90^{\\circ}-75^{\\circ}\\right)\\right) \\cdot\\left(\\tan 25^{\\circ} \\cdot \\cot \\left(90^{\\circ}-65^{\\circ}\\right)\\right) \\cdot \\times \\sqrt{3}$<\/p>\n\n\n\n

$\\left(\\tan 15^{\\circ} \\cdot \\cot 15^{\\circ}\\right) \\cdot\\left(\\tan 25^{\\circ} \\cdot \\cot 25^{\\circ}\\right) \\cdot \\times \\sqrt{3}$<\/p>\n\n\n\n

$=1 \\times 1 \\times \\sqrt{3}=\\sqrt{3}$ R.H.S<\/p>\n\n\n\n

(iii) $\\frac{2 \\sin ^{2} 63^{\\circ}+1+2 \\sin ^{2} 27^{\\circ}}{3 \\cos ^{2} 17^{\\circ}-2+3 \\cos ^{2} 73^{\\circ}}=3$<\/strong><\/p>\n\n\n\n

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u0947 \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f \u0964<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin 50^{\\circ}}{\\cos 40^{\\circ}}+\\frac{\\operatorname{cosec} 40^{\\circ}}{\\sec 50^{\\circ}}-4 \\cos 50^{\\circ} \\cdot \\operatorname{cosec} 40^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cos \\left(90^{\\circ}-50^{\\circ}\\right)}{\\cos 40^{\\circ}}+\\frac{\\sec \\left(90^{\\circ}-40^{\\circ}\\right)}{\\sec 50^{\\circ}}-4 \\sin \\left(90^{\\circ}-50^{\\circ}\\right) \\cdot \\operatorname{Cosec} 40^{\\circ}$<\/p>\n\n\n\n

$\\frac{\\cos 40^{\\circ}}{\\cos 40^{\\circ}}+\\frac{\\sec 50^{\\circ}}{\\sec 50^{\\circ}}-4 \\sin 40^{\\circ} \\cdot \\operatorname{Cosec} 40^{\\circ}$<\/p>\n\n\n\n

=1+1-4\u00d71<\/p>\n\n\n\n

=2-4=2<\/p>\n\n\n\n

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

$\\frac{\\cos ^{2} 20^{\\circ}+\\cos ^{2} 70^{\\circ}}{\\sin ^{2} 59^{\\circ}+\\sin ^{2} 31^{\\circ}}+\\sin 35^{\\circ} \\cdot \\sec 55^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin ^{2}\\left(90^{\\circ}-20^{\\circ}\\right)+\\cos ^{2} 70^{\\circ}}{\\cos ^{2}\\left(90^{\\circ}-59^{\\circ}\\right)+\\sin ^{2} 31^{\\circ}}+\\cos \\left(90^{\\circ}-35^{\\circ}\\right) \\cdot \\operatorname{Sec} 55^{\\circ}$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} 70^{\\circ}+\\cos ^{2} 70^{\\circ}}{\\cos ^{2} 31^{\\circ}+\\sin ^{2} 31^{\\circ}}+\\cos 55^{\\circ} \\cdot \\operatorname{Sec} 55^{\\circ}$<\/p>\n\n\n\n

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

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

$\\frac{\\tan 50^{\\circ}+\\sec 50^{\\circ}}{\\cot 40^{\\circ}+\\operatorname{cosec} 40^{\\circ}}+\\cos 40^{\\circ} \\cdot \\operatorname{cosec} 50^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cot \\left(90^{\\circ}-5^{\\circ}\\right)+\\operatorname{cose}\\left(90^{\\circ}-50^{\\circ}\\right)}{\\cot 40^{\\circ}+\\operatorname{cosec} 40^{\\circ}}+\\sin \\left(90^{\\circ}-40^{\\circ}\\right) \\cdot \\operatorname{cosec} 50^{\\circ}$<\/p>\n\n\n\n

$\\frac{\\cot 40^{\\circ}+\\operatorname{cosec} 40^{\\circ}}{\\cot 40^{\\circ}+\\operatorname{cosec} 40^{\\circ}}+\\sin 50^{\\circ} \\cdot \\operatorname{cosec} 50^{\\circ}$<\/p>\n\n\n\n

1+1=2<\/p>\n\n\n\n

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

$\\operatorname{cosec}\\left(65^{\\circ}+\\theta\\right)-\\sec \\left(25^{\\circ}-\\theta\\right)-\\tan \\left(55^{\\circ}-\\theta\\right)+\\cot \\left(35^{\\circ}+\\theta\\right)$<\/strong><\/p>\n\n\n\n

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

$\\sec \\left(90^{\\circ}-\\left(65^{\\circ}+\\theta\\right)\\right)-\\sec \\left(25^{\\circ}-\\theta\\right)-\\tan \\left(55^{\\circ}-\\theta\\right)+\\cot \\left(90^{\\circ}-\\left(35^{\\circ}+\\theta\\right)\\right)$<\/p>\n\n\n\n

$\\sec \\left(90^{\\circ}-65^{\\circ}-\\theta\\right)-\\sec \\left(25^{\\circ}-\\theta\\right)-\\tan \\left(55^{\\circ}-\\theta\\right)+\\cot \\left(90^{\\circ}-35^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

$\\sec \\left(25^{\\circ}-\\theta\\right)-\\sec \\left(25^{\\circ}-\\theta\\right)-\\tan \\left(55^{\\circ}-\\theta\\right)+\\cot \\left(55^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

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

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

$\\frac{\\cos 35^{\\circ}}{\\sin 55^{\\circ}}+\\frac{\\sin 11^{\\circ}}{\\cos 79^{\\circ}}-\\cos 28^{\\circ} \\cdot \\operatorname{cosec} 62^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin \\left(90^{\\circ}-3^{\\circ}\\right)}{\\sin 55^{\\circ}}+\\frac{\\cos \\left(90^{\\circ}-11^{\\circ}\\right)}{\\cos 79^{\\circ}}-\\sin \\left(90^{\\circ}-28^{\\circ}\\right) . \\operatorname{Cosec} 62^{\\circ}$<\/p>\n\n\n\n

$\\frac{\\sin 55^{\\circ}}{\\sin 55^{\\circ}}+\\frac{\\cos 79^{\\circ}}{\\cos 79^{\\circ}}-\\sin \\left(90^{\\circ}-28^{\\circ}\\right) \\cdot \\operatorname{Cosec} 62^{\\circ}$<\/p>\n\n\n\n

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

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

$\\frac{\\cos ^{2} 20^{\\circ}+\\cos ^{2} 70^{\\circ}}{\\sin ^{2} 59^{\\circ}+\\sin ^{2} 31^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin ^{2}\\left(90^{\\circ}-20^{\\circ}\\right)+\\cos ^{2} 70^{\\circ}}{\\cos ^{2}\\left(90^{\\circ}-59^{\\circ}\\right)+\\sin ^{2} 31^{\\circ}}$<\/p>\n\n\n\n

$\\frac{\\sin ^{2} 70^{\\circ}+\\cos ^{2} 70^{\\circ}}{\\cos ^{2} 31^{\\circ}+\\sin ^{2} 31^{\\circ}}$<\/p>\n\n\n\n

$\\frac{1}{1}=1$<\/p>\n\n\n\n

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

$\\operatorname{cosec}\\left(65^{\\circ}+, \\theta\\right)-\\sec \\left(25^{\\circ}-\\theta\\right)$<\/strong><\/p>\n\n\n\n

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

$\\sec \\left(90^{\\circ}-\\left(65^{\\circ}+\\theta\\right)\\right)-\\sec \\left(25^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

$\\sec \\left(90^{\\circ}-65^{\\circ}-\\theta\\right)-\\sec \\left(25^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

$\\sec \\left(25^{\\circ}-\\theta\\right)-\\sec \\left(25^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

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

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

$\\cos \\left(60^{\\circ}+\\theta\\right)-\\sin \\left(30^{\\circ}-\\theta\\right)$<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-\\left(60^{\\circ}+\\theta\\right)\\right)-\\sin \\left(30^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

$\\sin \\left(90^{\\circ}-60^{\\circ}-\\theta\\right)-\\sin \\left(30^{\\circ}-\\theta\\right)$<\/p>\n\n\n\n

$\\sin \\left(30^{\\circ}-\\theta\\right)-\\sin \\left(30^{\\circ}-\\theta\\right)=0$<\/p>\n\n\n\n

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

$\\sec 70^{\\circ} \\cdot \\sin 20^{\\circ}-\\cos 20^{\\circ} \\cdot \\operatorname{cosec} 70^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\operatorname{cosec}\\left(90^{\\circ}-70^{\\circ}\\right) \\cdot \\sin 20^{\\circ} \\cdot-\\sin \\left(90^{\\circ}-20^{\\circ}\\right) \\cdot \\operatorname{cosec} 70^{\\circ}$ $\\operatorname{cosec} 20^{\\circ} . \\sin 20^{\\circ} .-\\sin 70^{\\circ} . \\operatorname{cosec} 70^{\\circ}$ <\/p>\n\n\n\n

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

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

$\\left(\\sin 72^{\\circ}+\\cos 18^{\\circ}\\right)\\left(\\sin 72^{\\circ}-\\cos 18^{\\circ}\\right)$<\/strong><\/p>\n\n\n\n

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

$=\\sin ^{2} 72^{\\circ}-\\cos ^{2} 18^{\\circ}$<\/p>\n\n\n\n

$=\\cos ^{2}\\left(90^{\\circ}-72^{\\circ}\\right)-\\cos ^{2} 18^{\\circ}$<\/p>\n\n\n\n

$=\\cos ^{2} 18^{\\circ}-\\cos ^{2} 18^{\\circ}$<\/p>\n\n\n\n

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

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

$\\left(\\frac{\\sin 35^{\\circ}}{\\cos 55^{\\circ}}\\right)^{2}+\\left(\\frac{\\cos 55^{\\circ}}{\\sin 35^{\\circ}}\\right)-2 \\cos 60^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\cos \\left(90^{\\circ}-3^{\\circ}\\right)}{\\cos 55^{\\circ}}+\\frac{\\sin \\left(90^{\\circ}-55^{\\circ}\\right)}{\\sin 35^{\\circ}}-2 \\times \\frac{1}{2}$<\/p>\n\n\n\n

$\\frac{\\cos 55^{\\circ}}{\\cos 55^{\\circ}}+\\frac{\\sin 35^{\\circ}}{\\sin 35^{\\circ}}-1$<\/p>\n\n\n\n

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

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

$\\frac{\\cos 80^{\\circ}}{\\sin 10^{\\circ}}+\\cos 59^{\\circ} \\cdot \\operatorname{cosec} 31^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\cos \\left(90^{\\circ}-\\left(50^{\\circ}+\\theta\\right)\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)+\\tan 1^{\\circ} \\cdot \\tan 89^{\\circ} \\cdot \\tan 10^{\\circ} \\cdot \\tan 80^{\\circ} \\cdot \\tan 20^{\\circ} \\cdot \\tan 70^{\\circ}$<\/p>\n\n\n\n

$\\cos \\left(90^{\\circ}-50^{\\circ}-\\theta\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)+\\tan 1^{\\circ} \\cdot \\cot \\left(90^{\\circ}-89^{\\circ}\\right) \\cdot \\tan 10^{\\circ} \\cdot \\cot \\left(90^{\\circ}-80^{\\circ}\\right) \\cdot \\tan 20^{\\circ} \\cdot \\cot \\left(90^{\\circ}-70^{\\circ}\\right)$<\/p>\n\n\n\n

$\\cos \\left(40^{\\circ}-\\theta\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)+\\tan 1^{\\circ} \\cdot \\cot 1^{\\circ} \\cdot \\tan 10^{\\circ} \\cdot \\cot 10^{\\circ} \\cdot \\tan 20^{\\circ} \\cdot \\cot 20^{\\circ}$<\/p>\n\n\n\n

=0+1\u00d71\u00d71=1<\/p>\n\n\n\n

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

$\\sin \\left(50^{\\circ}+\\theta\\right)-\\cos \\left(40^{\\circ}-\\theta\\right)+\\tan 1^{\\circ} \\cdot \\tan 10^{\\circ} \\cdot \\tan 20^{\\circ} \\cdot \\tan 70^{\\circ} \\cdot \\tan 80^{\\circ} \\tan 89^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\operatorname{cosec}^{2}\\left(90^{\\circ}-10^{\\circ}\\right)-\\cot ^{2} 80^{\\circ}+\\frac{\\sin 15^{\\circ} \\times \\sin \\left(90^{\\circ}-75^{\\circ}\\right)+\\cos 15^{\\circ} \\times \\cos \\left(90^{\\circ}-75^{\\circ}\\right)}{\\cos \\theta \\times \\cos \\theta+\\sin \\theta \\times \\sin \\theta}$<\/p>\n\n\n\n

$\\operatorname{cosec}^{2} 80^{\\circ}-\\cot ^{2} 80^{\\circ}+\\frac{\\sin 15^{\\circ} \\times \\sin 15^{\\circ}+\\cos 15^{\\circ} \\times \\cos 15^{\\circ}}{\\cos ^{2} \\theta+\\sin ^{2} \\theta}$<\/p>\n\n\n\n

$1+\\frac{\\sin ^{2} 15^{\\circ}+\\cos ^{2} 15^{\\circ}}{\\cos ^{2} \\theta+\\sin ^{2} \\theta}$<\/p>\n\n\n\n

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

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

$\\sec ^{2} 10^{\\circ}-\\cot ^{2} 80^{\\circ}+\\frac{\\sin 15^{\\circ} \\cdot \\cos 75^{\\circ}+\\cos 15^{\\circ} \\cdot \\sin 75^{\\circ}}{\\cos \\theta \\sin \\left(90^{\\circ}-\\theta\\right)+\\sin \\theta \\cos \\left(90^{\\circ}-\\theta\\right)}$<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-\\left(90^{\\circ}-\\left(40^{\\circ}+\\theta\\right)\\right)-\\sin \\left(50^{\\circ}-\\theta\\right)+\\frac{\\sin ^{2}\\left(90^{\\circ}-40^{\\circ}\\right)+\\cos ^{2} 50^{\\circ}}{\\cos ^{2}\\left(90^{\\circ}-40^{\\circ}\\right)+\\sin ^{2} 50^{\\circ}}\\right.$<\/p>\n\n\n\n

$\\sin \\left(90^{\\circ}-40^{\\circ}-\\theta\\right)-\\sin \\left(50^{\\circ}-\\theta\\right)+\\frac{\\sin ^{2} 50^{\\circ}+\\cos ^{2} 50^{\\circ}}{\\cos ^{2} 50^{\\circ}+\\sin ^{2} 50^{\\circ}}$<\/p>\n\n\n\n

$\\sin \\left(50^{\\circ}-\\theta\\right)-\\left(50^{\\circ}-\\theta\\right)+1$<\/p>\n\n\n\n

$0+\\frac{1}{1}=1$<\/p>\n\n\n\n

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

$\\cos \\left(40^{\\circ}+\\theta\\right)-\\sin \\left(50^{\\circ}-\\theta\\right)+\\frac{\\cos ^{2} 40^{\\circ}+\\cos ^{2} 50^{\\circ}}{\\sin ^{2} 40^{\\circ}+\\sin ^{2} 50^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-\\left(90^{\\circ}-\\left(40^{\\circ}+\\theta\\right)\\right)-\\sin \\left(50^{\\circ}-\\theta\\right)+\\frac{\\sin ^{2}\\left(90^{\\circ}-40^{\\circ}\\right)+\\cos ^{2} 50^{\\circ}}{\\cos ^{2}\\left(90^{\\circ}-40^{\\circ}\\right)+\\sin ^{2} 50^{\\circ}}\\right.$<\/p>\n\n\n\n

$\\sin \\left(90^{\\circ}-40^{\\circ}-\\theta\\right)-\\sin \\left(50^{\\circ}-\\theta\\right)+\\frac{\\sin ^{2} 50^{\\circ}+\\cos ^{2} 50^{\\circ}}{\\cos ^{2} 50^{\\circ}+\\sin ^{2} 50^{\\circ}}$<\/p>\n\n\n\n

$\\sin \\left(50^{\\circ}-\\theta\\right)-\\left(50^{\\circ}-\\theta\\right)+1$<\/p>\n\n\n\n

$0+\\frac{1}{1}=1$<\/p>\n\n\n\n

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

$\\frac{\\cos 70^{\\circ}}{\\sin 20^{\\circ}}+\\frac{\\cos 55^{\\circ} \\cdot \\operatorname{cosec} 35^{\\circ}}{\\tan 5^{\\circ} \\cdot \\tan 25^{\\circ} \\cdot \\tan 45^{\\circ} \\cdot \\tan 65^{\\circ} \\cdot \\tan 85^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin \\left(90^{\\circ}-70^{\\circ}\\right)}{\\sin 20^{\\circ}}+\\frac{\\sin \\left(90^{\\circ}-55^{\\circ}\\right) \\times \\operatorname{cosec} 35^{\\circ}}{\\left(\\tan 5^{\\circ} \\times \\tan 85^{\\circ}\\right) \\times\\left(\\tan 25^{\\circ} \\times \\tan 65^{\\circ}\\right) \\times \\tan 45^{\\circ}}$<\/p>\n\n\n\n

$\\frac{\\sin 20^{\\circ}}{\\sin 20^{\\circ}}+\\frac{\\sin 35^{\\circ} \\times \\operatorname{cosec} 35^{\\circ}}{\\left(\\tan 5^{\\circ} \\times \\cot \\left(90^{\\circ}-85^{\\circ}\\right) \\times\\left(\\tan 25^{\\circ} \\times \\cot \\left(90^{\\circ}-65^{\\circ}\\right) \\tan 45^{\\circ}\\right.\\right.}$<\/p>\n\n\n\n

$1+\\frac{1}{\\left(\\tan 5^{\\circ} \\times \\cot 5^{\\circ}\\right)\\left(\\tan 25^{\\circ} \\times \\cot 25^{\\circ}\\right) \\times 1}$<\/p>\n\n\n\n

$1+\\frac{1}{1 \\times 1 \\times 1}=1+\\frac{1}{1}=1+1=2$<\/p>\n\n\n\n

Question 47<\/h4>\n\n\n\n

$\\left(\\frac{\\sin 27^{\\circ}}{\\cos 63^{\\circ}}\\right)^{2}+\\left(\\frac{\\cos 63^{\\circ}}{\\sin 27^{\\circ}}\\right)^{2}$<\/strong><\/p>\n\n\n\n

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

$\\left(\\frac{\\cos \\left(90^{\\circ}-27^{\\circ}\\right)}{\\cos 63^{\\circ}}\\right)^{2}+\\left(\\frac{\\sin \\left(90^{\\circ}-63^{\\circ}\\right)}{\\sin 27^{\\circ}}\\right)^{2}$<\/p>\n\n\n\n

$\\left(\\frac{\\cos 63^{\\circ}}{\\cos 63^{\\circ}}\\right)^{2}+\\left(\\frac{\\sin 27^{\\circ}}{\\sin 27^{\\circ}}\\right)^{2}$<\/p>\n\n\n\n

1+1=2<\/p>\n\n\n\n

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

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u0947 \u092e\u093e\u0928 \u091c\u094d\u091e\u093e\u0924 \u0915\u0940\u091c\u093f\u090f \u0964<\/strong><\/p>\n\n\n\n

(i) $\\frac{3 \\sin 5^{\\circ}}{\\cos 85^{\\circ}}+\\frac{2 \\cos 33^{\\circ}}{\\sin 57^{\\circ}}$<\/strong><\/p>\n\n\n\n

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

$\\frac{3 \\cos \\left(90^{\\circ}-5^{\\circ}\\right)}{\\cos 85^{\\circ}}+\\frac{2 \\sin \\left(90^{\\circ}-33^{\\circ}\\right)}{\\sin 57^{\\circ}}$<\/p>\n\n\n\n

$\\frac{3 \\cos 65^{\\circ}}{\\cos 85^{\\circ}}+\\frac{2 \\sin 57^{\\circ}}{\\sin 57^{\\circ}}$<\/p>\n\n\n\n

3\u00d71+2\u00d71=3+2<\/p>\n\n\n\n

=5<\/p>\n\n\n\n

(ii) $\\frac{\\cot 54^{\\circ}}{\\tan 36^{\\circ}}+\\frac{\\tan 20^{\\circ}}{\\cot 70^{\\circ}}-2$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\tan \\left(90^{\\circ}-54^{\\circ}\\right)}{\\tan 36^{\\circ}}+\\frac{\\cot \\left(90^{\\circ}-20^{\\circ}\\right)}{\\cot 70^{\\circ}}-2$<\/p>\n\n\n\n

$\\frac{\\tan 36^{\\circ}}{\\tan 36^{\\circ}}+\\frac{\\cot 76^{\\circ}}{\\cot 70^{\\circ}}-2$<\/p>\n\n\n\n

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

(iii) $\\frac{\\cos 80^{\\circ}}{\\sin 10^{\\circ}}+\\cos 59^{\\circ} \\operatorname{cosec} 31^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\frac{\\sin \\left(90^{\\circ}-80^{\\circ}\\right)}{\\sin 10^{\\circ}}+\\sin \\left(90^{\\circ}-59^{\\circ}\\right) \\times \\operatorname{cosec} 31^{\\circ}$<\/p>\n\n\n\n

$\\frac{\\sin 30^{\\circ}}{\\sin 10^{\\circ}}+\\sin 31^{\\circ} \\times \\operatorname{cosec} 31^{\\circ}$<\/p>\n\n\n\n

1+1=2<\/p>\n\n\n\n

(iv) $\\cos 38^{\\circ} \\cos 52^{\\circ}-\\sin 38^{\\circ} \\sin 52^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\sin \\left(90^{\\circ}-38^{\\circ}\\right) \\times \\cos 52^{\\circ}-\\cos \\left(90^{\\circ}-38^{\\circ}\\right) \\times \\sin 52^{\\circ}$<\/p>\n\n\n\n

$\\sin 52^{\\circ} \\times \\cos 52^{\\circ}-\\cos 52^{\\circ} \\times \\sin 52^{\\circ}$<\/p>\n\n\n\n

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

(v) $\\sec 41^{\\circ} \\sin 49^{\\circ}+\\operatorname{coss} 49^{\\circ} \\operatorname{cosec} 41^{\\circ}$<\/strong><\/p>\n\n\n\n

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

$\\operatorname{cosec}\\left(90^{\\circ}-41^{\\circ}\\right) \\times \\sin 49^{\\circ}+\\sin \\left(90^{\\circ}-49^{\\circ}\\right) \\times \\operatorname{cosec} 41^{\\circ}$<\/p>\n\n\n\n

$\\operatorname{cosec} 49^{\\circ} \\times \\sin 49^{\\circ}+\\sin 41^{\\circ} \\times \\operatorname{cosec} 41^{\\circ}$<\/p>\n\n\n\n

1+1=2<\/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":"

\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u094b 90\u00b0-\u03b8 \u0915\u0947 \u092a\u0942\u0930\u0915 \u0915\u094b\u0923 \u0915\u0940 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0940\u091c\u093f\u090f \u0964 (i) tan(90\u00b0-\u03b8) Sol : cot \u03b8 (ii) cos(90\u00b0-\u03b8) Sol : sin \u03b8 Question 3 \u0930\u093f\u0915\u094d\u0924 \u0938\u094d\u0925\u093e\u0928\u094b\u0902 \u0915\u0940 \u092a\u0930\u094d\u0924\u093f 0\u00b0 \u0914\u0930 90\u00b0 \u0915\u0947 \u092c\u0940\u091a \u0915\u0947 \u0915\u093f\u0938\u0940 \u0915\u094b\u0923 \u0938\u0947 \u0915\u0930\u0947\u0902\u0964 (i) $\\sin 70^{\\circ}=\\cos (\\ldots)$ Sol :cos(90\u00b0-70\u00b0)=20\u00b0 (ii) $\\sin 35^{\\circ}=\\cos (\\ldots)$ Sol : $\\cos \\left(90^{\\circ}-35^{\\circ}\\right)=55^{\\circ}$ (iii) $\\cos 48^{\\circ}=\\sin […]<\/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-625696","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-class-10","entry"],"yoast_head":"\nKC Sinha: Exercise 8.3 - 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=\"\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u094b 90\u00b0-\u03b8 \u0915\u0947 \u092a\u0942\u0930\u0915 \u0915\u094b\u0923 \u0915\u0940 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0940\u091c\u093f\u090f \u0964 (i) tan(90\u00b0-\u03b8) Sol : cot \u03b8 (ii) cos(90\u00b0-\u03b8) Sol :\" \/>\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-8-3-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 8.3 - 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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":"\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u094b 90\u00b0-\u03b8 \u0915\u0947 \u092a\u0942\u0930\u0915 \u0915\u094b\u0923 \u0915\u0940 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0940\u091c\u093f\u090f \u0964 (i) tan(90\u00b0-\u03b8) Sol : cot \u03b8 (ii) cos(90\u00b0-\u03b8) Sol :","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-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","og_locale":"en_US","og_type":"article","og_title":"KC Sinha: Exercise 8.3 - 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":"\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u094b 90\u00b0-\u03b8 \u0915\u0947 \u092a\u0942\u0930\u0915 \u0915\u094b\u0923 \u0915\u0940 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0940\u091c\u093f\u090f \u0964 (i) tan(90\u00b0-\u03b8) Sol : cot \u03b8 (ii) cos(90\u00b0-\u03b8) Sol :","og_url":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-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":"2023-09-08T01:17:34+00:00","article_modified_time":"2023-09-08T01:36:37+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":"11 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#article","isPartOf":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-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 8.3 – 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":"2023-09-08T01:17:34+00:00","dateModified":"2023-09-08T01:36:37+00:00","mainEntityOfPage":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/"},"wordCount":3923,"publisher":{"@id":"https:\/\/www.indcareer.com\/schools\/#organization"},"image":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-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-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","url":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/","name":"KC Sinha: Exercise 8.3 - 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-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/#primaryimage"},"image":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-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":"2023-09-08T01:17:34+00:00","dateModified":"2023-09-08T01:36:37+00:00","description":"\u0928\u093f\u092e\u094d\u0928\u0932\u093f\u0916\u093f\u0924 \u0915\u094b 90\u00b0-\u03b8 \u0915\u0947 \u092a\u0942\u0930\u0915 \u0915\u094b\u0923 \u0915\u0940 \u0924\u094d\u0930\u093f\u0915\u094b\u0923\u092e\u093f\u0924\u0940\u092f \u0905\u0928\u0941\u092a\u093e\u0924 \u0915\u0947 \u0930\u0942\u092a \u092e\u0947\u0902 \u0935\u094d\u092f\u0915\u094d\u0924 \u0915\u0940\u091c\u093f\u090f \u0964 (i) tan(90\u00b0-\u03b8) Sol : cot \u03b8 (ii) cos(90\u00b0-\u03b8) Sol :","breadcrumb":{"@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-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-8-3-mathematics-solution-class-10-chapter-8-trikonmitiy-anupat-evam-sarvasamikaye\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.indcareer.com\/schools\/kc-sinha-exercise-8-3-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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