KC Sinha Mathematics Solution Class 11 Chapter 6 Trigonometric Functions Exercise 6.2

 Exercise 6.2

Page no 6.31

Type 1

Question 1

Express each of the following products into sums or difference of sines and cosines.

(i) $2 \cos 3 \theta \cdot \sin 2 \theta$

(ii) $2 \sin 5 \theta \cdot \cos 3 \theta$

(iii) $\cos 9 \theta \cdot \cos 4 \theta$

(iv) $\sin 75^{\circ} \cdot \cos 15^{\circ}$

Page no 6.32

Question 2

Prove that $\sin 25^{\circ} \cos 115^{\circ}=\frac{1}{2}\left(\sin 40^{\circ}-1\right)$

Question 3

Prove that

(i) $\sin 20^{\circ} \sin 40^{\circ} \sin 60^{\circ} \sin 80^{\circ}=\frac{3}{16}$

(ii) $\cos 20^{\circ} \cos 40^{\circ} \cos 80^{\circ}=\frac{1}{8}$

Question 4

Prove that

(i) $\cos 10^{\circ} \cos 30^{\circ} \cos 50^{\circ} \cos 70^{\circ}=\frac{3}{16}$

(ii) $\tan 20^{\circ} \tan 30^{\circ} \tan 40^{\circ} \tan 80^{\circ}=1$

Question 5

Prove that

(i) $4 \cos \theta \cos \left(\frac{\pi}{3}+\theta\right) \cos \left(\frac{\pi}{3}-\theta\right)=\cos 3 \theta$

(ii) $\cos 2 x \cos \frac{x}{2}-\cos 3 x \cos \frac{9 x}{2}=\sin 5 x \sin \frac{5 x}{2}$

Question 6

Show that : $\sin (B-C) \cos (A-D)+\sin (C-A)$ $\cos (B-D)$ $+\sin (A-B) \cos (C-D)=0$

Question 7

If $\alpha+\beta=90^{\circ}$, show that the maximum value of $\cos \alpha, \cos \beta$ is $\frac{1}{2}$

Question 8

Prove that $\tan \theta \tan \left(60^{\circ}-\theta\right) \cdot \tan \left(60^{\circ}+\theta\right)=\tan 3 \theta$

Question 9

If $\cos \alpha=\frac{1}{\sqrt{2}}, \sin \beta=\frac{1}{\sqrt{3}}$, show that $\tan \frac{\alpha+\beta}{2} \cot \frac{\alpha-\beta}{2}=5+2 \sqrt{6}$ or $5-2 \sqrt{6}$

[Hint : Since $\cos \alpha=\frac{1}{\sqrt{2}}(+$ ve), therefore, $\alpha$ lies in the Ist or 4 th gradient and hence $\sin \alpha=\frac{1}{\sqrt{2}}$ or $-\frac{1}{\sqrt{2}} 1$

Type 2

Question 10

Express each of the following as product of sines and cosines :

(i) $\cos 9 \theta+\cos 3 \theta$

(ii) $\sin 2 \theta+\cos 4 \theta$

(iii) $\cos 12 \theta-\cos 4 \theta$

(iv) $\sin 9 \theta+\sin 5 \theta$

Question 11

Prove that

(i) $\sin 65^{\circ}+\cos 65^{\circ}=\sqrt{2} \cos 20^{\circ}$

(ii) $\sin 47^{\circ}+\cos 77^{\circ}=\cos 17^{\circ}$

Question 12

Prove that

(i) $\frac{\cos 7 x+\cos 5 x}{\sin 7 x-\sin 5 x}=\cot x$

(ii) $\frac{\cos 9 x-\cos 5 x}{\sin 17 x-\sin 3 x}=\frac{-\sin 2 x}{\cos 10 x}$

(iii) $\frac{\sin x+\sin 3 x}{\cos x+\cos 3 x}=\tan 2 x$

(iv) $\cot 4 x(\sin 5 x+\sin 3 x)=\cot x(\sin 5 x-\sin 3 x)$

(v) $\frac{\sin x-\sin 3 x}{\sin ^{2} x-\cos ^{2} x}=2 \sin x$

[Hint : $\left.\sin ^{2} A-\cos ^{2} B=\cos (A+B) \cos (A-B)\right]$

Question 13

Prove that

(i) $\frac{\cos 10^{\circ}-\sin 10^{\circ}}{\cos 10^{\circ}+\sin 10^{\circ}}=\tan 35^{\circ}$

(ii) $\cos 80^{\circ}+\cos 40^{\circ}-\cos 20^{\circ}=0$

(iii) $\sin 10^{\circ}+\sin 20^{\circ}+\sin 40^{\circ}+\sin 50^{\circ}=\sin 70^{\circ}+\sin 80^{\circ}$

(iv) $\cos 20^{\circ}+\cos 100^{\circ}+\cos 140^{\circ}=0$

Question 14

Prove that

(i) $\cos \frac{\pi}{5}+\cos \frac{2 \pi}{5}+\cos \frac{6 \pi}{5}+\cos \frac{7 \pi}{5}=0$

(ii) $\cos \frac{\pi}{12}-\sin \frac{\pi}{12}=\frac{1}{\sqrt{2}}$

(iii) $\sin \frac{5 \pi}{18}-\cos \frac{4 \pi}{9}=\sqrt{3} \sin \frac{\pi}{9}$

(iv) $\cos \left(\frac{3 \pi}{4}+x\right)-\cos \left(\frac{3 \pi}{4}-x\right)=-\sqrt{2} \sin x$

(v) $\cos \left(\frac{\pi}{4}+x\right)+\cos \left(\frac{\pi}{4}-x\right)=\sqrt{2} \cos x$

Question 15

Prove that $\cos \alpha+\cos \beta+\cos \gamma+\cos (\alpha+\beta+\gamma)$

$=4 \cos \frac{\alpha+\beta}{2} \cdot \cos \frac{\beta+\gamma}{2} \cdot \cos \frac{\gamma+\alpha}{2}$

Question 16

Prove that

(i) $\frac{\sin x-\sin y}{\cos x+\cos y}=\tan \frac{x-y}{2}$

(ii) $\frac{\sin x+\sin y}{\cos x+\cos y}=\tan \frac{x+y}{2}$

(iii) $\frac{\sin x+\sin y}{\sin x-\sin y}=\tan \left(\frac{x+y}{2}\right) \cdot \cot \left(\frac{x-y}{2}\right)$

Question 17

Prove that

(i) $\sin 3 x+\sin 2 x-\sin x=4 \sin x \cos \frac{x}{2} \cos \frac{3 x}{2}$

(ii) $\sin x+\sin 3 x+\sin 5 x+\sin 7 x=4 \cos x \cos 2 x \sin 4 x$

(iii) $\frac{\sin \theta+\sin 3 \theta+\sin 5 \theta}{\cos \theta+\cos 3 \theta+\cos 5 \theta}=\tan 3 \theta$

(iv) $\frac{\cos 4 \theta+\cos 3 \theta+\cos 2 \theta}{\sin 4 \theta+\sin 3 \theta+\sin 2 \theta}=\cot 3 \theta$

(v) $\frac{\sin \theta+2 \sin 3 \theta+\sin 5 \theta}{\sin 3 \theta+2 \sin 5 \theta+\sin 7 \theta}=\frac{\sin 3 \theta}{\sin 5 \theta}$

(vi) $\frac{\cos 3 \theta+2 \cos 5 \theta+\cos 7 \theta}{\cos \theta+2 \cos 3 \theta+\cos 5 \theta}=\cos 2 \theta-\sin 2 \theta$, $\tan 3 \theta$

(vii) $\frac{\sin 5 \theta-2 \sin 3 \theta+\sin \theta}{\cos 5 \theta-\cos \theta}=\tan \theta$

Page no 6.34

Question 18

If $\operatorname{cosec} A+\sec A=\operatorname{cosec} B+\sec B$, prove that $\tan A \cdot \tan B=\cot \frac{A+B}{2}$

Question 19

Prove that

(i) $(\cos \alpha-\cos \beta)^{2}+(\sin \alpha-\sin \beta)^{2}=4 \sin ^{2}\left(\frac{\alpha-\beta}{2}\right)$

(ii) $\sin \alpha+\sin \beta+\sin \gamma-\sin (\alpha+\beta+\gamma)=4 \sin \left(\frac{\alpha+\beta}{2}\right) \cdot \sin \left(\frac{\beta+\gamma}{2}\right) \cdot \sin \left(\frac{\gamma+\alpha}{2}\right)$

Type 3

Question 20

If $\frac{\cos (A-B)}{\cos (A+B)}+\frac{\cos (C+D)}{\cos (C-D)}=0$, prove that $\tan A \cdot \tan B \cdot \tan C \cdot \tan D=-1$

Question 21

If $\sin 2 A=\lambda \sin 2 B$, prove that $\frac{\tan (A+B)}{\tan (A-B)}=\frac{\lambda+1}{\lambda-1}$

Question 22

If $\cos (\alpha+\beta) \sin (\gamma+\delta)=\cos (\alpha-\beta) \cdot \sin (\gamma-\delta)$, prove that $\cot \alpha \cdot \cot \beta \cdot \cot \gamma=\cot \delta$

Question 23

If $y \sin \phi=x \sin (2 \theta+0)$ show that $(x+y) \cot (\theta+0)=(y-x) \cot \theta$