Exercise 7.2
Page no - 7.31
Type 1
Question 1
Prove that $\frac{2 \sin \theta-\sin 2 \theta}{2 \sin \theta+\sin 2 \theta}=\tan ^{2} \frac{\theta}{2}$
Question 2
Prove that $\cot \frac{\theta}{2}-\tan \frac{\theta}{2}=2 \cot \theta$
Question 3
Prove that $\frac{1+\sin \theta}{1-\sin \theta}=\tan ^{2}\left(\frac{\pi}{4}+\frac{\theta}{2}\right)$
Question 4
Prove that $\sec \theta+\tan \theta=\tan \left(\frac{\pi}{4}+\frac{\theta}{2}\right)$
Question 5
Prove that $\frac{\sin \alpha+\sin \beta-\sin (\alpha+\beta)}{\sin \alpha+\sin \beta+\sin (\alpha+\beta)}=\tan \frac{\alpha}{2} \tan \frac{\beta}{2}$
Question 6
Prove that $\tan \left(\frac{\pi}{4}-\frac{A}{2}\right)=\sec A-\tan A=\sqrt{\frac{1-\sin A}{1+\sin A}}$
Question 7
Prove that $\operatorname{cosec}\left(\frac{\pi}{4}+\frac{\theta}{2}\right) \operatorname{cosec}\left(\frac{\pi}{4}-\frac{\theta}{2}\right)=2 \sec \theta$
Question 8
Prove that
(i) $(\cos A-\cos B)^{2}+(\sin A-\sin B)^{2}=4 \sin ^{2} \frac{A-B}{2}$
(ii) $(\cos x+\cos y)^{2}+(\sin x-\sin y)^{2}=4 \cos ^{2} \frac{x+y}{2}$
Question 9
(i) Prove that $\cos ^{2} \frac{\pi}{8}+\cos ^{2} \frac{3 \pi}{8}+\cos ^{2} \frac{5 \pi}{8}+\cos ^{2} \frac{7 \pi}{8}=2$
(ii) Prove that $\sin ^{4} \frac{\pi}{8}+\sin ^{4} \frac{3 \pi}{8}+\sin ^{4} \frac{5 \pi}{8}+\sin ^{4} \frac{7 \pi}{8}=\frac{3}{2}$
Question 10
(i) If $\tan x=\frac{3}{4}, \pi<x<\frac{3 \pi}{2}$, find the values of $\sin \frac{x}{2}$ and $\cos \frac{x}{2}$
(ii) If $\sin x=\frac{1}{4}, \frac{\pi}{2}<x<\pi$, find the values of $\cos \frac{x}{2}$ and $\tan \frac{x}{2}$
(iii) If $\cos x=-\frac{1}{3}, \pi<x<\frac{3 \pi}{2}$, find the values of $\sin \frac{x}{2}$ and $\tan \frac{x}{2}$
Question 11
Prove that
$\left(1+\cos \frac{\pi}{8}\right)\left(1+\cos \frac{3 \pi}{8}\right)\left(1+\cos \frac{5 \pi}{8}\right)\left(1+\cos \frac{7 \pi}{8}\right)=\frac{1}{8}$
Type 2
Question 12
Show that $\cot 142 \frac{1^{\circ}}{2}=\sqrt{2}+\sqrt{3}-2-\sqrt{6}$
[Hint : $\left.\cot 142 \frac{1^{\circ}}{2}=\cot \left(180^{\circ}-37 \frac{1^{\circ}}{2}\right)=-\cot 37 \frac{1^{\circ}}{2}\right]$
Page no - 7.32
Question 13
Prove that $\sin ^{2} 48^{\circ}-\cos ^{2} 12^{\circ}=-\frac{\sqrt{5+1}}{8}$
Question 14
Prove that $4\left(\sin 24^{\circ}+\cos 6^{\circ}\right)=\sqrt{3}+\sqrt{15}$
[Hint : L.H.S. $\left.=4\left(\sin 24^{\circ}+\sin 84^{\circ}\right)\right]$
Question 15
Prove that $\cot 6^{\circ} \cot 42^{\circ} \cot 66^{\circ} \cot 78^{\circ}=1$
Question 16
Prove that $\tan 12^{\circ} \tan 24^{\circ} \tan 48^{\circ} \tan 84^{\circ}=1$
Question 17
Prove that $\sin 6^{\circ} \sin 42^{\circ} \sin 66^{\circ} \sin 78^{\circ}=\frac{1}{16}$
Question 18
Prove that $\sin \frac{\pi}{5} \sin \frac{2 \pi}{5} \sin \frac{3 \pi}{5} \sin \frac{4 \pi}{5}=\frac{5}{16}$
Question 19
Prove that $\cos 36^{\circ} \cos 72^{\circ} \cos 108^{\circ} \cos 144^{\circ}=\frac{1}{16}$
Question 20
Prove that $\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{3 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{5 \pi}{15} \cos \frac{6 \pi}{15} \cos \frac{7 \pi}{15}=\frac{1}{2^{7}}$
Question 21
Show that $\cos \frac{\pi}{65} \cos \frac{2 \pi}{65} \cos \frac{4 \pi}{65} \cos \frac{8 \pi}{65} \cos \frac{16 \pi}{65} \cos \frac{32 \pi}{65}=\frac{1}{64}$
Type 3
Question 22
If $\tan \frac{\theta}{2}=\sqrt{\frac{1-e}{1+e}} \tan \frac{\varphi}{2}$, prove that $\cos \varphi=\frac{\cos \theta-e}{1-e \cos \theta}$
Question 23
If $\sin \alpha+\sin \beta=a$ and $\cos \alpha+\cos \beta=b$, prove that
(i) $\sin (\alpha+\beta)=\frac{2 a b}{a^{2}+b^{2}}$
(ii) $\cos (\alpha-\beta)=\frac{1}{2}\left(a^{2}+b^{2}-2\right)$
Question 24
If $\alpha$ and $\beta$ be the two different roots of equation $a \cos \theta+b \sin \theta=c$, prove that
(i) $\tan (\alpha+\beta)=\frac{2 a b}{a^{2}-b^{2}}$
(ii) $\cos (\alpha+\beta)=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$
Question 25
If $\cos \alpha+\cos \beta=\frac{1}{3}$ and $\sin \alpha+\sin \beta=\frac{1}{4}$. prove that $\cos \frac{\alpha-\beta}{2}=\pm \frac{5}{24}$
Question 26
If $2 \tan \frac{\alpha}{2}=\tan \frac{\beta}{2}$, prove that $\cos \alpha=\frac{3+5 \cos \beta}{5+3 \cos \beta}$
Question 27
If $\sin \alpha=\frac{4}{5}$ and $\cos \beta=\frac{5}{13}$, prove that one value of $\cos \frac{\alpha-\beta}{2}=\frac{8}{\sqrt{65}}$
Question 28
If $\sec (\varphi+\alpha)+\sec (\varphi-\alpha)=2 \sec \varphi$. prove that $\cos \varphi=\pm \sqrt{2} \cos \frac{\alpha}{2}$
Question 29
If $\cos \theta=\frac{\cos \alpha \cos \beta}{1-\sin \alpha \sin \beta}$, prove that one value of
$\tan \frac{\theta}{2}=\frac{\tan \frac{\alpha}{2}-\tan \frac{\beta}{2}}{1-\tan \frac{\alpha}{2} \tan\frac{\beta}{2}}$
Page no 7.33
Question 30
If $\tan \alpha=\frac{\sin \theta \sin \varphi}{\cos \theta+\cos \varphi}$, prove that one of the values of $\tan \frac{\alpha}{2}$ is $\tan \frac{\theta}{2} \tan \frac{\varphi}{2}$,
Question 31
If $\cos \theta=\frac{\cos \alpha+\cos \beta}{1+\cos \alpha \cos \beta}$, prove that one of the values of $\tan \frac{\theta}{2}$ is $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}$
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