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 $(x+y) \cot (\theta+0)=(y-x) \cot \theta$
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