KC Sinha: Exercise 8.3 - Mathematics Solution Class 10 Chapter 8 त्रिकोणमितीय अनुपात एवम सर्वसमिकाए

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निम्नलिखित को 90°-θ के पूरक कोण की त्रिकोणमितीय अनुपात के रूप में व्यक्त कीजिए ।

(i) tan(90°-θ)

Sol :

cot θ

(ii) cos(90°-θ)

Sol :

sin θ

Question 3

रिक्त स्थानों की पर्ति 0° और 90° के बीच के किसी कोण से करें।

(i) $\sin 70^{\circ}=\cos (\ldots)$

Sol :
cos(90°-70°)=20°

(ii) $\sin 35^{\circ}=\cos (\ldots)$

Sol :

$\cos \left(90^{\circ}-35^{\circ}\right)=55^{\circ}$

(iii) $\cos 48^{\circ}=\sin (\ldots)$

Sol :

$\sin \left(90^{\circ}-48^{\circ}\right)=42^{\circ}$

(iv) $\cos 70^{\circ}=\sin (\ldots)$

Sol :

$\sin \left(90^{\circ}-70^{\circ}\right)=20^{\circ}$

(v) $\cos 50^{\circ}=\sin (\ldots)$

Sol :

$\sin \left(90^{\circ}-50^{\circ}\right)=40^{\circ}$

(vi) $\sec 32^{\circ}=\operatorname{cosec}(\ldots)$

Sol :

$\operatorname{cosec}\left(90^{\circ}-32^{\circ}\right)=58^{\circ}$

TYPE II: कोण θ के त्रिकोणामितीय अनुपात दिए रहने पर न्यूनकोण θ के पूएक कोण के त्रिकोणमितीय अनुपातों को ज्ञात करने पर आधारित प्रश्न :

Question 4

यदि $\mathrm{A}+\mathrm{B}=90^{\circ}$, तो पूरक कोण $\mathrm{A}$ या B के उपयुक्त त्रिकोणमिताय अनुपात से रिक्त स्थानों की पूर्ति कीजिए।

(i) sin A=....

Sol :

$=\sin \left(90^{\circ}-B\right)=\cos B$

(ii) cos B=...

Sol :

$=\cos \left(90^{\circ}-A\right)=\cos A$

(iii) sec A=...

Sol :

$=\sec \left(90^{\circ}-\mathrm{B}\right)=\operatorname{cosec} \mathrm{B}$

(iv) tan B=...

Sol :

$=\tan \left(90^{\circ}-\mathrm{A}\right)=\cot \mathrm{A}$

(v) cosec B=...

Sol :

$=\operatorname{cosec}\left(90^{\circ}-\mathrm{A}\right)=\sec \mathrm{A}$

(vi) cot A=...

$=\cot \left(90^{\circ}-B\right)=\tan B$

Question 5

(i) यदि $\sin 37^{\circ}=a$, तब $\cos 53^{\circ}$ का मान a के पदों में व्यक्त करें।

Sol :

$\sin 37^{\circ}=a$

$\cos \left(90^{\circ}-37^{\circ}\right)=a$

$\cos 53^{\circ}=\mathrm{a}$

(ii) यदि $\cos 47^{\circ}=a$, तब $\sin 43^{\circ}$ का मान a के पदों में व्यक्त करें।

Sol :

$\cos 47^{\circ}=a$

$\sin \left(90^{\circ}-47^{\circ}\right)=a$

$\sin 43^{\circ}=a$

(iii) यदि $\sin 52^{\circ}=a$, तब $\cos 38^{\circ}$ का मान a के पदों में व्यक्त करें।

Sol :

$\sin 52^{\circ}=a$

$\cos \left(90^{\circ}-52^{\circ}\right)=a$

$\cos 38^{\circ}=a$

(iv) यदि $\sin 56^{\circ}=x$, तब $\sin 34^{\circ}$ का मान x के पदों में व्यक्त करें।

Sol :

$\sin 56^{\circ}=x$ 

$\cos \left(90^{\circ}-56^{\circ}\right)=x$ 

$\cos 34^{\circ}=x$

Type III: पूरक कोणों के त्रिकोणभितीय अनुपातों से सम्बर्द्ध व्यंजकों के मानों पर आयाएित प्रश्न:

Question 6

निम्नलिखित के मान ज्ञात करें।

(i) $\frac{\cos 59^{\circ}}{\sin 31^{\circ}}$

Sol :

$\frac{\sin \left(90^{\circ}-59\right)}{\sin 31^{\circ}}=\frac{\sin 31^{\circ}}{\sin 31^{\circ}}=1$

(ii) $\frac{\cos 53^{\circ}}{\sin 37^{\circ}}$

Sol :

$\frac{\sin \left(90^{\circ}-53^{\circ}\right)}{\sin 37^{\circ}}=\frac{\sin 37^{\circ}}{\sin 37^{\circ}}=1$

(iii) $\frac{\sin 20^{\circ}}{\cos 70^{\circ}}$

Sol :

$\frac{\cos \left(90^{\circ}-20^{\circ}\right)}{\cos 70^{\circ}}=\frac{\cos 70^{\circ}}{\cos 70^{\circ}}=1$

(iv) $\frac{\sqrt{2} \sin 22^{\circ}}{\cos 68^{\circ}}$

Sol :

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

(v) $\frac{\sin 10^{\circ}}{\cos 80^{\circ}}$

Sol :

$\frac{\cos \left(90^{\circ}-10^{\circ}\right)}{\cos 80^{\circ}}=\frac{\cos 80^{\circ}}{\cos 80^{\circ}}=1$

(vi) $\frac{\sin 27^{\circ}}{\cos 63^{\circ}}$

Sol :

$\frac{\cos \left(90^{\circ}-27^{\circ}\right)}{\cos 63^{\circ}}=\frac{\cos 63^{\circ}}{\cos 63^{\circ}}=1$

(vii) $\frac{\sqrt{3} \cos 65^{\circ}}{\sin 25^{\circ}}$

Sol :

$\frac{\sqrt{3} \sin \left(90^{\circ}-65^{\circ}\right)}{\sin 25^{\circ}}$

$=\frac{\sqrt{3} \sin 25^{\circ}}{\sin 25^{\circ}}=\sqrt{3} \times 1=\sqrt{3}$

(viii) $\frac{\cos 29^{\circ}}{\sin 61^{\circ}}$

Sol :

$\frac{\sin \left(90^{\circ}-29^{\circ}\right)}{\sin 61^{\circ}}=\frac{\sin 61^{\circ}}{\sin 61^{\circ}}=1$

(ix) $\sin 54^{\circ}-\cos 36^{\circ}$

Sol :

$\cos \left(90^{\circ}-54^{\circ}\right)-\cos 36^{\circ}$

$\cos 36^{\circ}-\cos 36^{\circ}=0$

(x) $\frac{\tan 80^{\circ}}{\cot 10^{\circ}}$

Sol :

$\frac{\cot \left(90^{\circ}-80^{\circ}\right)}{\cot 10^{\circ}}=\frac{\cot 10^{\circ}}{\operatorname{cot} 10^{\circ}}=1$

(xi) $ \operatorname{cosec} 31^{\circ}-\sec 59^{\circ}$

Sol :

$\sec \left(90^{\circ}-31^{\circ}\right)-\sec 59^{\circ}$

$\sec 59^{\circ}-\sec 59^{\circ}=0$

(xii) $\frac{\sin 18^{\circ}}{\cos 72^{\circ}}$

Sol :

$\frac{\cos \left(90^{\circ}-18^{\circ}\right)}{\cos 72^{\circ}}=\frac{\cos 72^{\circ}}{\cos 72^{\circ}}=1$

(xiii) $\frac{\tan 65^{\circ}}{\cos 25^{\circ}}$

Sol :

$\frac{\cot \left(90^{\circ}-65^{\circ}\right)}{\cot 25^{\circ}}=\frac{\cot 25^{\circ}}{\cot 25^{\circ}}=1$

Question 7

रिक्त स्थानों को भरें-

(i) यदि $\sin 50^{\circ}=0.7660$, तो $\cos 40^{\circ}=\ldots \ldots . .$

Sol :

$\cos \left(90^{\circ}-50^{\circ}\right)=0.7660$

$\cos 40^{\circ}=0.7660$

(ii) यदि $\cos 44^{\circ}=0.7193$, तो $\sin 46^{\circ}=$.

Sol :

$\sin \left(90^{\circ}-44^{\circ}\right)=0.7193$

$\sin 46^{\circ}=0.7193$

(iii) $\sin 50^{\circ}+\cos 40^{\circ}=2 \sin (\ldots \ldots . .)$

Sol :

$\sin 50^{\circ}+\cos 40^{\circ}$

$\sin 50^{\circ}+\sin \left(90^{\circ}+40^{\circ}\right)$

$\sin 50^{\circ}+\sin 50^{\circ}$

$2 \sin 50^{\circ}$

(iv) $\frac{\sin 70^{\circ}}{\cos 20^{\circ}}$ का मान $\ldots \ldots$ है

Sol :

$\frac{\cos \left(90^{\circ}-70^{\circ}\right)}{\cos 20^{\circ}}=\frac{\cos 20^{\circ}}{\cos 20^{\circ}}=1$

Question 8

(i) यदि $\mathrm{A}+\mathrm{B}=90^{\circ}$, तब $\cos \mathrm{B}$ को $\mathrm{A}$ के सरलतम त्रिकोणमितीय अनुपात में व्यक्त करें।

Sol :

$\cos B=\cos \left(90^{\circ}-A\right)=\sin A$

(ii) यदि $\mathrm{X}+\mathrm{Y}=90^{\circ}$, तब $\cos \mathrm{X}$ को $\mathrm{Y}$ के सरलतम त्रिकोणमितीय अनुपात में व्यक्त करें।

Sol :

$\cos X=\cos \left(90^{\circ}-Y\right)=\sin Y$

Question 9

(i) यदि $\mathrm{A}+\mathrm{B}=90^{\circ}, \sin \mathrm{A}=a, \sin \mathrm{B}=b$, तो सिद्ध कीजिए कि-

(a) $a^{2}+b^{2}=1$

Sol :

sin A=a, sin B=b

दोनों को वर्ग करके जोड़ने पर

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

(b) $\tan \mathrm{A}=\frac{a}{b}$

Sol :

sin A=a, sin B=b

भाग देने पर

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

(ii) दिखायें कि $\sin \left(50^{\circ}+\theta\right)-\cos \left(40^{\circ}-\theta\right)=0$.

Sol :

$\sin \left(50^{\circ}+\theta\right)-\cos \left(40^{\circ}-\theta\right)=0$

$\cos \left[90^{\circ}-\left(50^{\circ}+\theta\right)-\cos \left(40^{\circ}-\theta\right)\right.$

$\cos \left(90^{\circ}-50^{\circ}-\theta\right)-\cos \left(40^{\circ}-\theta\right)$

$\cos \left(40^{\circ}-\theta\right)-\cos \left(40^{\circ}-\theta\right)$=0

Question 10

सिद्ध कीजिए कि $\frac{\cos \theta}{\sin \left(90^{\circ}-\theta\right)}+\frac{\sin \theta}{\cos \left(90^{\circ}-\theta\right)}=2$

Sol :

LHS

$\frac{\cos \theta}{\cos \theta}+\frac{\sin \theta}{\sin \theta}=2$

1+1=2

2=2 proved

Question 11

किसी $\triangle \mathrm{ABC}$ में सिद्ध कीजिए कि-

(a) $\sin \frac{\mathrm{B}+\mathrm{C}}{2}=\cos \frac{\mathrm{A}}{2}$

Sol :

त्रिभुज के तीनों कोणो का मान $180^{\circ}$ होता है

$A+B+C=180^{\circ}$

$B+C=180^{\circ}-A$

LHS

$\sin \frac{B+C}{2}=\cos \frac{A}{2}$

$\sin \left(\frac{180^{\circ}-\mathrm{A}}{2}\right)=\cos \frac{\mathrm{A}}{2}$

$\sin \left(\frac{180^{\circ}}{2}-\frac{A}{2}\right)=\cos \frac{A}{2}$

$\sin \left(90^{\circ}-\frac{A}{2}\right)=\cos \frac{A}{2}$

$\cos \frac{A}{2}=\cos \frac{A}{2}$

(b) $\tan \frac{\mathrm{B}+\mathrm{C}}{2}=\cot \frac{\mathrm{A}}{2}$

Sol :

त्रिभुज के तीनों कोणो का मान $180^{\circ}$ होता है

$\begin{aligned}&A+B+C=180^{\circ} \\&B+C=180^{\circ}-A\end{aligned}$

LHS

$\tan \frac{B+C}{2}=\cot \frac{A}{2}$

$\tan \left(\frac{180^{\circ}-A}{2}\right)=\cot \frac{A}{2}$

$\tan \left(\frac{180^{\circ}}{2}-\frac{A}{2}\right)=\cot \frac{A}{2}$

$\tan \left(90^{\circ}-\frac{A}{2}\right)=\cot \frac{A}{2}$

$\cot \frac{A}{2}=\cot \frac{A}{2}$

(c) $\cos \frac{\mathrm{A}+\mathrm{B}}{2}=\sin \frac{\mathrm{C}}{2}$

Sol :

त्रिभुज के तीनों कोणो का मान $180^{\circ}$ होता है

$\begin{aligned}&A+B+C=180^{\circ} \\&A+B=180^{\circ}-A\end{aligned}$

L.H.S

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

$\cos \frac{A+B}{2}=\sin \frac{A}{2}$

$\cos \left(\frac{180^{\circ}-C}{2}\right)=\sin \frac{C}{2}$

$\cos \left(\frac{180^{\circ}}{2}-\frac{C}{2}\right)=\sin \frac{C}{2}$

$\cos \left(90^{\circ}-\frac{\mathrm{C}}{2}\right)=\sin \frac{\mathrm{C}}{2}$

$\sin \frac{\mathrm{C}}{2}=\sin \frac{\mathrm{C}}{2}$

Question 12

(i) यदि $\sin 3 \mathrm{~A}=\cos \left(\mathrm{A}-26^{\circ}\right)$, जहाँ $3 \mathrm{~A}$ एक न्यूनकोण है तब $\mathrm{A}$ का मान ज्ञात कीजिए ।

Sol :

$\cos \left(90^{\circ}-3 A\right)=\cos \left(A-26^{\circ}\right)$

$90^{\circ}-3 A=A-26^{\circ}$

$90^{\circ}+26^{\circ}=A+3 A$

$116^{\circ}=4 A$

$A=\frac{116}{4}$

$A=29^{\circ}$

(ii) यदि $\cos \left(2 \theta+54^{\circ}\right)=\sin \theta$. जहाँ $\left(2 \theta+54^{\circ}\right)$ एक न्यूनकोण है तब $\theta$ का मान ज्ञात कीजिए ।

Sol :

$\cos \left(2 \theta+54^{\circ}\right)=\cos \left(90^{\circ}-\theta\right)$

$2 \theta+54^{\circ}=90^{\circ}-\theta$

$2 \theta+\theta=90^{\circ}-54^{\circ}$

$3 \theta=36^{\circ}$

$\theta=\frac{36^{\circ}}{3}$

$\theta=12^{\circ}$

(iii) यदि $\tan 3 \theta=\cot \left(\theta+18^{\circ}\right)$, जहाँ $3 \theta$ और $\theta+18^{\circ}$ न्यूनकोण हैं, तो $\theta$ का मान ज्ञात कीजिए ।

Sol :

$\cot =\left(90^{\circ}-3 \theta\right)=\cot \left(\theta+18^{\circ}\right)$

$90^{\circ}-3 \theta=\theta+18^{\circ}$

$-3 \theta-\theta=18^{\circ}-90^{\circ}$

$-4 \theta=-72^{\circ}$

$\theta=\frac{72}{4}$

$\theta=18^{\circ}$

(iv) यदि $\sec 5 \theta=\operatorname{cosec}\left(\theta-36^{\circ}\right)$, जहाँ 5θ एक न्यूनकोण है, तो θ का मान निकालिए ।

Sol :

$\operatorname{cosec}\left(90^{\circ}-5 \theta\right)=\operatorname{cosec}\left(\theta-36^{\circ}\right)$

$90^{\circ}-5 \theta=\theta-36^{\circ}$

$90^{\circ}+36^{\circ}=\theta+5 \theta$

$126^{\circ}=6 \theta$

$6 \theta=126^{\circ}$

$\theta=\frac{126}{6}$

$\theta=21^{\circ}$

सिद्ध कीजिए कि :

Question 13

$\sin 70^{\circ} \cdot \sec 20^{\circ}=1$

Sol :

L.H.S

$\sin 70^{\circ} . \operatorname{Sec} 20^{\circ}$

$\sin 70^{\circ} \cdot \operatorname{cosec}\left(90^{\circ}-20^{\circ}\right)$

$\sin 70^{\circ} \cdot \operatorname{cosec} 70^{\circ}=1$ R.H.S

Question 14

$\sin \left(90^{\circ}-\theta\right) \tan \theta=\sin \theta$

Sol :

L.H.S

$\sin \left(90^{\circ}-\theta\right) \cdot \tan \theta$

$\cos \theta \times \frac{\sin \theta}{\cos \theta}$

$=\sin \theta$ R.H.S

Question 15

$\tan 63^{\circ} \cdot \tan 27^{\circ}=1$

Sol :

L.H.S

$\tan 63^{\circ} . \tan 27$

$\tan 63^{\circ} \cdot \cot \left(90^{\circ}-27^{\circ}\right)$

$\tan 63^{\circ} \cdot \cot 63^{\circ}$

=1 R.H.S

Question 16

$\frac{\sin \left(90^{\circ}-\theta\right) \sin \theta}{\tan \theta}-1=-\sin ^{2} \theta$

Sol :

L.H.S

$

\begin{aligned}

&\frac{\sin \left(90^{\circ}-\theta\right) \sin \theta}{\tan \theta}-1 \\

&\frac{\cos \theta \cdot \sin \theta}{\frac{\sin \theta}{\cos \theta}}-1 \\

&\cos ^{2}-1 \\

&-\sin ^{2} \theta \text { R.H.S }

\end{aligned}

$

Question 17

$\sin 55^{\circ} \cdot \cos 48^{\circ}=\cos 35^{\circ} \cdot \sin 42^{\circ}$

Sol :

L.H.S

$\sin 55^{\circ} \cdot \cos 48^{\circ}$

$\cos \left(90^{\circ}-55^{\circ}\right) \cdot \sin \left(90^{\circ}-48^{\circ}\right)$

$\cos 35^{\circ} . \operatorname{Sin} 42^{\circ} \quad$ R.H.S

Question 18

$\sin 25^{\circ} \cdot \sin 65^{\circ}=\cos 25^{\circ} \cdot \cos 65^{\circ}$

Sol :

L.H.S

$\sin 25^{\circ} . \operatorname{Sin} 65^{\circ}$

$\cos \left(90^{\circ}-25^{\circ}\right) \cdot \cos \left(90^{\circ}-65^{\circ}\right)$

$\cos 25^{\circ} \cdot \cos 65^{\circ} \quad$ R.H.S

Question 19

$\sin 54^{\circ}+\cos 67^{\circ}=\sin 23^{\circ}+\cos 36^{\circ}$

Sol :

L.H.S

$\sin 54^{\circ} \cdot \cos 67^{\circ}$

$\cos \left(90^{\circ}-54^{\circ}\right) \cdot \sin \left(90^{\circ}-67^{\circ}\right)$

$\sin 23^{\circ} \cdot \cos 36^{\circ} \quad$ R.H.S

Question 20

$\cos 27^{\circ}+\sin 51^{\circ}=\sin 63^{\circ}+\cos 39^{\circ}$

Sol :

L.H.S

$\cos 27^{\circ} \cdot \sin 51^{\circ}$

$\sin \left(90^{\circ}-27^{\circ}\right) \cdot \cos \left(90^{\circ}-51^{\circ}\right)$

$\sin 63^{\circ} \cdot \cos 39^{\circ} \quad$ R.H.S

Question 21

$\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ}=1$

Sol :

L.H.S

$

\begin{aligned}

&\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ} \\

&\sin ^{2} 40^{\circ}+\cos ^{2}\left(90^{\circ}-50^{\circ}\right) \\

&\sin ^{2} 40^{\circ}+\cos ^{2} 40^{\circ} \\

&=1 \quad \text { R.H.S }

\end{aligned}

$

Question 22

$\sin ^{2} 29^{\circ}+\sin ^{2} 61^{\circ}=1$

Sol :

L.H.S

$

\begin{aligned}

&\sin ^{2} 29^{\circ}+\sin ^{2} 61^{\circ} \\

&\sin ^{2} 29^{\circ}+\cos ^{2}\left(90^{\circ}-61^{\circ}\right)

\end{aligned}

$

$\sin ^{2} 29^{\circ}+\cos ^{2} 29^{\circ}$

=1 R.H.S

Question 23

$\sin \theta \cdot \cos \left(90^{\circ}-\theta\right)+\cos \theta \sin \left(90^{\circ}-\theta\right)=1$

Sol :

L.H.S

$

\begin{aligned}

&\sin \theta \cdot \cos \left(90^{\circ}-\theta\right)+\cos \theta \cdot \sin \left(90^{\circ}-\theta\right) \\

&\sin \theta \times \sin \theta+\cos \theta \times \cos \theta \\

&\sin ^{2} \theta+\cos ^{2} \theta \\

&=1 \quad \text { R.H.S }

\end{aligned}

$

Question 24

$\cos \theta \cdot \cos \left(90^{\circ}-\theta\right)-\sin \theta \cdot \sin \left(90^{\circ}-\theta\right)=0$

Sol :

L.H.S

$

\begin{aligned}

&\cos \theta \cdot \cos \left(90^{\circ}-\theta\right)-\sin \theta \cdot \sin \left(90^{\circ}-\theta\right) \\

&\cos \theta \cdot \sin \theta-\sin \theta \cdot \cos \theta \\

&=0 \quad \text { R.H.S }

\end{aligned}

$

Question 25

$\sin 42^{\circ} \cdot \cos 48^{\circ}+\cos 42^{\circ} \cdot \sin 48^{\circ}=1$

Sol :

L.H.S

$\sin 42^{\circ} \cdot \cos 48^{\circ}+\cos 42^{\circ} \cdot \sin 48^{\circ}$

$\sin 42^{\circ} \cdot \sin \left(90^{\circ}-48^{\circ}\right)+\cos 42^{\circ} \cdot \cos \left(90^{\circ}-48^{\circ}\right)$

$\sin 42^{\circ} \cdot \sin 42^{\circ}+\cos 42^{\circ} \cdot \cos 42^{\circ}$

=1 R.H.S

Question 26

$\frac{\cos 20^{\circ}}{\sin 70^{\circ}}+\frac{\cos \theta}{\sin \left(90^{\circ}-\theta\right)}=2$

Sol :

L.H.S

$

\begin{aligned}

&\frac{\cos 20^{\circ}}{\sin 70^{\circ}}+\frac{\cos \theta}{\sin \left(90^{\circ}-\theta\right)} \\

&\frac{\cos 20^{\circ}}{\sin 70^{\circ}}+\frac{\cos \theta}{\cos \theta)} \\

&\frac{\sin \left(90^{\circ}-20^{\circ}\right)}{\sin 70^{\circ}}+1 \\

&\frac{\sin 70^{\circ}}{\sin 70^{\circ}}+1 \end{aligned}$

Question 27

$\sin ^{2} 85^{\circ}+\sin ^{2} 5^{\circ}+\sin ^{2} 6 ?^{\circ}+\sin ^{2}, 23^{\circ}=2$

Sol :

L.H.S

$

\begin{aligned}

&\sin ^{2} 85^{\circ}+\sin ^{2} 5^{\circ}+\sin ^{2} 67^{\circ}+\sin ^{2} 23^{\circ} \\

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

&\sin ^{2} 85^{\circ}+\cos ^{2} 85^{\circ}+\sin ^{2} 67^{\circ}+\cos ^{2} 67^{\circ}

\end{aligned}

$

1+1=2 RHS

Question 28

$\tan 9^{\circ} \cdot \tan 27^{\circ} \cdot \tan 45^{\circ} \cdot \tan 63^{\circ} \cdot \tan 81^{\circ}=1$

Sol :

L.H.S

$\tan 9^{\circ} \cdot \tan 27^{\circ} \cdot \tan 45^{\circ} \cdot \tan 63^{\circ} \cdot \tan 81^{\circ}$

$\left(\tan 9^{\circ} \cdot \tan 81^{\circ}\right) \cdot\left(\tan 27^{\circ} \cdot \tan 63^{\circ}\right) \cdot \tan 45^{\circ}$

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

$\left(\tan 9^{\circ} \cdot \cot 9^{\circ}\right) \cdot\left(\tan 27^{\circ} \cdot \cot 27^{\circ}\right) \cdot \times 1$

=1×1=1 RHS

Question 29

$\sin 9^{\circ} \cdot \sin 27^{\circ} \cdot \sin 63^{\circ} \cdot \sin ^{\circ} 81^{\circ}=\cos 9^{\circ} \cdot \cos 27^{\circ} \cdot \cos 63^{\circ} \cdot \cos 81^{\circ}$

Sol :

L.H.S

$\sin 9^{\circ} \cdot \sin 27^{\circ} \cdot \sin 63^{\circ} \cdot \sin 81^{\circ}$

$\cos \left(90^{\circ}-9^{\circ}\right) \cdot \cos \left(90^{\circ}-27^{\circ}\right) \cdot \cos \left(90^{\circ}-63^{\circ}\right) \cdot \cos \left(90^{\circ}-81^{\circ}\right)$

$\cos 81^{\circ} \cdot \cos 63^{\circ} \cdot \cos 27^{\circ} \cdot \cos 9^{\circ}$

$\cos 9^{\circ} \cdot \cos 27^{\circ} \cdot \cos 63^{\circ} \cdot \cos 81^{\circ}$

R.H.S

Question 30

(i) $\tan 7^{\circ} \cdot \tan 23^{\circ}, \tan 60^{\circ} \cdot \tan 67^{\circ} \cdot \tan 83^{\circ}=\sqrt{3}$

Sol :

L.H.S

$\tan 7^{\circ} \cdot \tan 23^{\circ} \cdot \tan 60^{\circ} \cdot \tan 67^{\circ} \cdot \tan 83^{\circ}$

$\left(\tan 7^{\circ} \cdot \tan 83^{\circ}\right) \cdot\left(\tan 23^{\circ} \cdot \tan 67^{\circ}\right) \cdot \tan 60^{\circ}$

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

$\left(\tan 7^{\circ} \cdot \cot 7^{\circ}\right) \cdot\left(\tan 23^{\circ} \cdot \cot 23^{\circ}\right) \cdot \times \sqrt{3}$

$=1 \times 1 \times \sqrt{3}=\sqrt{3}$ R.H.S

(ii) $\tan 15^{\circ} \tan 25^{\circ} \tan 60^{\circ} \tan 65^{\circ} \tan 75^{\circ}=\sqrt{3}$

Sol :

L.H.S

$\tan 15^{\circ} \cdot \tan 25^{\circ} \cdot \tan 60^{\circ} \cdot \tan 65^{\circ} \cdot \tan 75^{\circ}$

$\left(\tan 15^{\circ} \cdot \tan 75^{\circ}\right) \cdot\left(\tan 25^{\circ} \cdot \tan 65^{\circ}\right) \cdot \tan 60^{\circ}$

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

$\left(\tan 15^{\circ} \cdot \cot 15^{\circ}\right) \cdot\left(\tan 25^{\circ} \cdot \cot 25^{\circ}\right) \cdot \times \sqrt{3}$

$=1 \times 1 \times \sqrt{3}=\sqrt{3}$ R.H.S

(iii) $\frac{2 \sin ^{2} 63^{\circ}+1+2 \sin ^{2} 27^{\circ}}{3 \cos ^{2} 17^{\circ}-2+3 \cos ^{2} 73^{\circ}}=3$

निम्नलिखित के मान ज्ञात कीजिए ।

Question 31

$\frac{\sin 50^{\circ}}{\cos 40^{\circ}}+\frac{\operatorname{cosec} 40^{\circ}}{\sec 50^{\circ}}-4 \cos 50^{\circ} \cdot \operatorname{cosec} 40^{\circ}$

Sol :

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

$\frac{\cos 40^{\circ}}{\cos 40^{\circ}}+\frac{\sec 50^{\circ}}{\sec 50^{\circ}}-4 \sin 40^{\circ} \cdot \operatorname{Cosec} 40^{\circ}$

=1+1-4×1

=2-4=2

Question 32

$\frac{\cos ^{2} 20^{\circ}+\cos ^{2} 70^{\circ}}{\sin ^{2} 59^{\circ}+\sin ^{2} 31^{\circ}}+\sin 35^{\circ} \cdot \sec 55^{\circ}$

Sol :

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

$\frac{\sin ^{2} 70^{\circ}+\cos ^{2} 70^{\circ}}{\cos ^{2} 31^{\circ}+\sin ^{2} 31^{\circ}}+\cos 55^{\circ} \cdot \operatorname{Sec} 55^{\circ}$

$\frac{1}{1}+1=2$

Question 33

$\frac{\tan 50^{\circ}+\sec 50^{\circ}}{\cot 40^{\circ}+\operatorname{cosec} 40^{\circ}}+\cos 40^{\circ} \cdot \operatorname{cosec} 50^{\circ}$

Sol :

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

$\frac{\cot 40^{\circ}+\operatorname{cosec} 40^{\circ}}{\cot 40^{\circ}+\operatorname{cosec} 40^{\circ}}+\sin 50^{\circ} \cdot \operatorname{cosec} 50^{\circ}$

1+1=2

Question 34

$\operatorname{cosec}\left(65^{\circ}+\theta\right)-\sec \left(25^{\circ}-\theta\right)-\tan \left(55^{\circ}-\theta\right)+\cot \left(35^{\circ}+\theta\right)$

Sol :

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

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

$\sec \left(25^{\circ}-\theta\right)-\sec \left(25^{\circ}-\theta\right)-\tan \left(55^{\circ}-\theta\right)+\cot \left(55^{\circ}-\theta\right)$

=0

Question 35

$\frac{\cos 35^{\circ}}{\sin 55^{\circ}}+\frac{\sin 11^{\circ}}{\cos 79^{\circ}}-\cos 28^{\circ} \cdot \operatorname{cosec} 62^{\circ}$

Sol :

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

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

1+1-1=2-1=1

Question 36

$\frac{\cos ^{2} 20^{\circ}+\cos ^{2} 70^{\circ}}{\sin ^{2} 59^{\circ}+\sin ^{2} 31^{\circ}}$

Sol :

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

$\frac{\sin ^{2} 70^{\circ}+\cos ^{2} 70^{\circ}}{\cos ^{2} 31^{\circ}+\sin ^{2} 31^{\circ}}$

$\frac{1}{1}=1$

Question 37

$\operatorname{cosec}\left(65^{\circ}+, \theta\right)-\sec \left(25^{\circ}-\theta\right)$

Sol :

$\sec \left(90^{\circ}-\left(65^{\circ}+\theta\right)\right)-\sec \left(25^{\circ}-\theta\right)$

$\sec \left(90^{\circ}-65^{\circ}-\theta\right)-\sec \left(25^{\circ}-\theta\right)$

$\sec \left(25^{\circ}-\theta\right)-\sec \left(25^{\circ}-\theta\right)$

=0

Question 38

$\cos \left(60^{\circ}+\theta\right)-\sin \left(30^{\circ}-\theta\right)$

Sol :

$\sin \left(90^{\circ}-\left(60^{\circ}+\theta\right)\right)-\sin \left(30^{\circ}-\theta\right)$

$\sin \left(90^{\circ}-60^{\circ}-\theta\right)-\sin \left(30^{\circ}-\theta\right)$

$\sin \left(30^{\circ}-\theta\right)-\sin \left(30^{\circ}-\theta\right)=0$

Question 39

$\sec 70^{\circ} \cdot \sin 20^{\circ}-\cos 20^{\circ} \cdot \operatorname{cosec} 70^{\circ}$

Sol :

$\operatorname{cosec}\left(90^{\circ}-70^{\circ}\right) \cdot \sin 20^{\circ} \cdot-\sin \left(90^{\circ}-20^{\circ}\right) \cdot \operatorname{cosec} 70^{\circ}$ $\operatorname{cosec} 20^{\circ} . \sin 20^{\circ} .-\sin 70^{\circ} . \operatorname{cosec} 70^{\circ}$ 

1-1=0

Question 40

$\left(\sin 72^{\circ}+\cos 18^{\circ}\right)\left(\sin 72^{\circ}-\cos 18^{\circ}\right)$

Sol :

$=\sin ^{2} 72^{\circ}-\cos ^{2} 18^{\circ}$

$=\cos ^{2}\left(90^{\circ}-72^{\circ}\right)-\cos ^{2} 18^{\circ}$

$=\cos ^{2} 18^{\circ}-\cos ^{2} 18^{\circ}$

=0

Question 41

$\left(\frac{\sin 35^{\circ}}{\cos 55^{\circ}}\right)^{2}+\left(\frac{\cos 55^{\circ}}{\sin 35^{\circ}}\right)-2 \cos 60^{\circ}$

Sol :

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

$\frac{\cos 55^{\circ}}{\cos 55^{\circ}}+\frac{\sin 35^{\circ}}{\sin 35^{\circ}}-1$

1+1-1=2-1=1

Question 42

$\frac{\cos 80^{\circ}}{\sin 10^{\circ}}+\cos 59^{\circ} \cdot \operatorname{cosec} 31^{\circ}$

Sol :

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

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

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

=0+1×1×1=1

Question 43

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

Sol :

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

$\operatorname{cosec}^{2} 80^{\circ}-\cot ^{2} 80^{\circ}+\frac{\sin 15^{\circ} \times \sin 15^{\circ}+\cos 15^{\circ} \times \cos 15^{\circ}}{\cos ^{2} \theta+\sin ^{2} \theta}$

$1+\frac{\sin ^{2} 15^{\circ}+\cos ^{2} 15^{\circ}}{\cos ^{2} \theta+\sin ^{2} \theta}$

$1+\frac{1}{1}=1+1=2$

Question 44

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

Sol :

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

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

$\sin \left(50^{\circ}-\theta\right)-\left(50^{\circ}-\theta\right)+1$

$0+\frac{1}{1}=1$

Question 45

$\cos \left(40^{\circ}+\theta\right)-\sin \left(50^{\circ}-\theta\right)+\frac{\cos ^{2} 40^{\circ}+\cos ^{2} 50^{\circ}}{\sin ^{2} 40^{\circ}+\sin ^{2} 50^{\circ}}$

Sol :

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

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

$\sin \left(50^{\circ}-\theta\right)-\left(50^{\circ}-\theta\right)+1$

$0+\frac{1}{1}=1$

Question 46

$\frac{\cos 70^{\circ}}{\sin 20^{\circ}}+\frac{\cos 55^{\circ} \cdot \operatorname{cosec} 35^{\circ}}{\tan 5^{\circ} \cdot \tan 25^{\circ} \cdot \tan 45^{\circ} \cdot \tan 65^{\circ} \cdot \tan 85^{\circ}}$

Sol :

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

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

$1+\frac{1}{\left(\tan 5^{\circ} \times \cot 5^{\circ}\right)\left(\tan 25^{\circ} \times \cot 25^{\circ}\right) \times 1}$

$1+\frac{1}{1 \times 1 \times 1}=1+\frac{1}{1}=1+1=2$

Question 47

$\left(\frac{\sin 27^{\circ}}{\cos 63^{\circ}}\right)^{2}+\left(\frac{\cos 63^{\circ}}{\sin 27^{\circ}}\right)^{2}$

Sol :

$\left(\frac{\cos \left(90^{\circ}-27^{\circ}\right)}{\cos 63^{\circ}}\right)^{2}+\left(\frac{\sin \left(90^{\circ}-63^{\circ}\right)}{\sin 27^{\circ}}\right)^{2}$

$\left(\frac{\cos 63^{\circ}}{\cos 63^{\circ}}\right)^{2}+\left(\frac{\sin 27^{\circ}}{\sin 27^{\circ}}\right)^{2}$

1+1=2

Question 48

निम्नलिखित के मान ज्ञात कीजिए ।

(i) $\frac{3 \sin 5^{\circ}}{\cos 85^{\circ}}+\frac{2 \cos 33^{\circ}}{\sin 57^{\circ}}$

Sol :

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

$\frac{3 \cos 65^{\circ}}{\cos 85^{\circ}}+\frac{2 \sin 57^{\circ}}{\sin 57^{\circ}}$

3×1+2×1=3+2

=5

(ii) $\frac{\cot 54^{\circ}}{\tan 36^{\circ}}+\frac{\tan 20^{\circ}}{\cot 70^{\circ}}-2$

Sol :

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

$\frac{\tan 36^{\circ}}{\tan 36^{\circ}}+\frac{\cot 76^{\circ}}{\cot 70^{\circ}}-2$

=1+1-2=0

(iii) $\frac{\cos 80^{\circ}}{\sin 10^{\circ}}+\cos 59^{\circ} \operatorname{cosec} 31^{\circ}$

Sol :

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

$\frac{\sin 30^{\circ}}{\sin 10^{\circ}}+\sin 31^{\circ} \times \operatorname{cosec} 31^{\circ}$

1+1=2

(iv) $\cos 38^{\circ} \cos 52^{\circ}-\sin 38^{\circ} \sin 52^{\circ}$

Sol :

$\sin \left(90^{\circ}-38^{\circ}\right) \times \cos 52^{\circ}-\cos \left(90^{\circ}-38^{\circ}\right) \times \sin 52^{\circ}$

$\sin 52^{\circ} \times \cos 52^{\circ}-\cos 52^{\circ} \times \sin 52^{\circ}$

1-1=0

(v) $\sec 41^{\circ} \sin 49^{\circ}+\operatorname{coss} 49^{\circ} \operatorname{cosec} 41^{\circ}$

Sol :

$\operatorname{cosec}\left(90^{\circ}-41^{\circ}\right) \times \sin 49^{\circ}+\sin \left(90^{\circ}-49^{\circ}\right) \times \operatorname{cosec} 41^{\circ}$

$\operatorname{cosec} 49^{\circ} \times \sin 49^{\circ}+\sin 41^{\circ} \times \operatorname{cosec} 41^{\circ}$

1+1=2