Exercise 7.1
Page no 7.12
Type 1
Question 1
If $\tan \theta=\frac{a}{b}$, where $0<\theta<\frac{\pi}{4}$ and $b>a>0$, find the values of $\sin 2 \theta, \cos 2 \theta$ and $\tan 2 \theta$.
Question 2
(i) If $\tan \theta=-\frac{3}{4}$ and $\frac{\pi}{2}<\theta<\pi$.
Find the values of $\sin \theta, \cos \theta$ and $\cot \theta$.
Question 3
Prove that $\cot \theta-\tan \theta=2 \cot 2 \theta$
Question 4
$1-\tan ^{2}\left(\frac{\pi}{4}-A\right)$
Prove that $\frac{\left(\frac{\pi}{4}-A\right)}{1+\tan ^{2}\left(\frac{\pi}{4}-A\right)}=\sin 2 A$
Question 5
Prove that $\frac{\cos \theta-\sin \theta}{\cos \theta+\sin \theta}=\sec 2 \theta-\tan 2 \theta$
Question 6
Prove that $\tan \left(\frac{\pi}{4}+\theta\right)-\tan \left(\frac{\pi}{4}-\theta\right)=2 \tan 2 \theta$
Question 7
If $\cos \theta=\frac{1}{2}\left(a+\frac{1}{a}\right)$, show that $\cos 2 \theta=\frac{1}{2}\left(a^{2}+\frac{1}{a^{2}}\right)$
Question 8
Prove that $\cos ^{2} \theta+\sin ^{2} \theta \cos 2 \beta=\cos ^{2} \beta+\sin ^{2} \beta \cos 2 \theta$
Question 9
Prove that $1+\tan \theta \tan 2 \theta=\sec 2 \theta$
Question 10
Prove that $\frac{1+\sin 2 A-\cos 2 A}{1+\sin 2 A+\cos 2 A}=\tan A$
Question 11
Prove that $\frac{1+\sin 2 \theta}{1-\sin 2 \theta}=\left(\frac{1+\tan \theta}{1-\tan \theta}\right)^{2}$
Question 12
Show that $\frac{1}{\sin 10^{\circ}}-\frac{\sqrt{3}}{\cos 10^{\circ}}=4$
Question 13
Prove that $\operatorname{cosec} A-2 \cot 2 A \cos A=2 \sin A$
Page no 7.13
Question 14
Prove that $\cot ^{2} A-\tan ^{2} A=4 \cot 2 A \operatorname{cosec} 2 A$
Question 15
Prove that $\frac{1+\sin 2 A}{\cos 2 A}=\frac{\cos A+\sin A}{\cos A-\sin A}=\tan \left(\frac{\pi}{4}+A\right)$
Question 16
Prove that $\cos ^{6} A-\sin ^{6} A=\cos 2 A\left(1-\frac{1}{4} \sin ^{2} 2 A\right)$
Question 17
Prove that $\cos ^{2} \theta+\cos ^{2}\left(\frac{\pi}{3}+\theta\right)+\cos ^{2}\left(\frac{\pi}{3}-\theta\right)=\frac{3}{2}$
Question 18
Prove that
(i) $\frac{2 \cos 2^{n} \theta+1}{2 \cos \theta+1}=(2 \cos \theta-1)(2 \cos 2 \theta-1)\left(2 \cos 2^{2} \theta-1\right) \ldots$ $\left(2 \cos 2^{n-1} \theta-1\right)$
(ii) $\frac{\tan 2^{n} \theta}{\tan \theta}=(1+\sec 2 \theta)\left(1+\sec 2^{2} \theta\right)\left(1+\sec 2^{3} \theta\right) \ldots\left(1+\sec 2^{n} \theta\right)$
Question 19
Prove that $\sin 2 x+2 \sin 4 x+\sin 6 x=4 \cos ^{2} x \sin 4 x$
Question 20
Prove that $\frac{\sin 2^{n} \theta}{\sin \theta}=2^{n} \cos \theta \cos 2 \theta \cos 2^{2} \theta \ldots \cos 2^{n-1} \theta$
Question 21
Show that $3(\sin x-\cos x)^{4}+6(\sin x+\cos x)^{2}+4\left(\sin ^{6} x+\cos ^{6} x\right)=13$
Question 22
Show that $2\left(\sin ^{6} x+\cos ^{6} x\right)-3\left(\sin ^{4} x+\cos ^{4} x\right)+1=0$
Question 23
Show that $\cos ^{2} \theta+\cos ^{2}(\alpha+\theta)-2 \cos \alpha \cos \theta \cos (\alpha+\theta)$ is independent of $\theta$.
Type 2
Question 24
Prove that $4\left(\cos ^{3} 10^{\circ}+\sin ^{3} 20^{\circ}\right)=3\left(\cos 10^{\circ}+\sin 20^{\circ}\right)$
Question 25
Prove that $\sin \theta \cos ^{3} \theta-\cos \theta \sin ^{3} \theta=\frac{1}{4} \sin 4 \theta$
Question 26
Prove that $\cos ^{3} \theta \cdot \sin 3 \theta+\sin ^{3} \theta \cdot \cos 3 \theta=\frac{3}{4} \sin 4 \theta$
Question 28
Prove that $4 \sin \theta \sin \left(\theta+\frac{\pi}{3}\right) \sin \left(\theta+\frac{2 \pi}{3}\right)=\sin 3 \theta$
Question 29
Prove that $\cot \theta+\cot \left(60^{\circ}+\theta\right)+\cot \left(120^{\circ}+\theta\right)=3 \cot 3 \theta$
Type 3
Question 30
Prove that $\cos 4 x=1-8 \sin ^{2} x \cos ^{2} x$
Question 31
Prove that $\sin 5 \theta=5 \sin \theta-20 \sin ^{3} \theta+16 \sin ^{5} \theta$
Question 32
Prove that, $\cos 6 \theta=32 \cos ^{6} \theta-48 \cos ^{4} \theta+18 \cos ^{4} \theta-1$
Question 33
Prove that
$\cos 4 \theta-\cos 4 \alpha=8(\cos \theta-\cos \alpha)(\cos \theta+\cos \alpha)(\cos \theta-\sin \alpha)(\cos\theta+\sin \alpha)$
Question 34
Prove that $\tan 4 \theta=\frac{4 \tan \theta-4 \tan ^{3} \theta}{1-6 \tan ^{2} \theta+\tan ^{4} \theta}$
Page no 7.14
Question 35
If $\tan x=\frac{b}{a}$, prove that $a \cos 2 x+b \sin 2 x=a$
Question 36
If $\tan ^{2} \theta=1+2 \tan ^{2} \phi$, prove that $\cos 2 \phi=1+2 \cos 2 \theta$
Question 37
If $\alpha$ and $\beta$ are acute angles and $\cos 2 \alpha=\frac{3 \cos 2 \beta-1}{3-\cos 2 \beta}$
Prove that $\tan \alpha=\sqrt{2} \tan \beta$
Question 38
If $\tan \beta=3 \tan \alpha$, prove that $\tan (\alpha+\beta)=\frac{2 \sin 2 \beta}{1+2 \cos 2 \beta}$
Question 39
If $x \sin \alpha=y \cos \alpha$, prove that $\frac{x}{\sec 2 \alpha}+\frac{y}{\operatorname{cosec} 2 \beta}=x$
Question 40
If $\tan \theta=\sec 2 \alpha$, prove that $\sin 2 \theta=\frac{1-\tan ^{4} \alpha}{1+\tan ^{4} \alpha}$
Question 41
If $\alpha=\frac{\pi}{3} \cdot$ prove that $\cos \alpha \cdot \cos 2 \alpha \cdot \cos 3 \alpha \cdot \cos 4 \alpha \cdot \cos 5 \alpha \cos 6 \alpha=\frac{1}{16}$
Question 42
If $\alpha=\frac{\pi}{15}$, prove that $\cos 2 \alpha \cos 4 \alpha \cos 8 \alpha \cos 14 \alpha=\frac{1}{16}$
Question 43
If $\tan A \tan B=\sqrt{\frac{a-b}{a+b}}$, prove that
$(a-b \cos 2 A)(a-b \cos 2 B)=a^{2}-b^{2}$
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