Exercise 6.1
Page no - 6.18
Type 1
Question 1
Find the value of :
(i) $\cos 210^{\circ}$
(ii) $\sin 15^{\circ}$
(iii) $\sin 75^{\circ}$
(iv) $\tan 15^{\circ}$
(v) $\tan 75^{\circ}$
(vi) $\tan \left(-330^{\circ}\right)$
(vii) $\cos 1395^{\circ}$
(viii) $\tan \left(\frac{11 \pi}{12}\right)$
(ix) $\tan \left(\frac{13 \pi}{12}\right)$
Question 2
Find the value of the following :
(i) $\sin 105^{\circ}+\cos 105^{\circ}$
(ii) $\sin 300^{\circ} \operatorname{cosec} 1050^{\circ}-\tan \left(-120^{\circ}\right)$
(iii) $\sin 690^{\circ} \cos 930^{\circ}+\tan \left(-765^{\circ}\right) \operatorname{cosec}\left(-1170^{\circ}\right)$
(iv) $\cot ^{2} \frac{\pi}{6}+\operatorname{cosec} \frac{5 \pi}{6}+3 \tan ^{2} \frac{\pi}{6}$
(v) $3 \sin \frac{\pi}{6} \sec \frac{\pi}{3}-4 \sin \frac{5 \pi}{6} \cot \frac{\pi}{4}$
(vi) $\sin ^{2} \frac{\pi}{6}+\cos ^{2} \frac{\pi}{3}-\tan ^{2} \frac{\pi}{4}$
Question 3
Prove that $\sin 75^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}$
Question 4
Find $\tan 15^{\circ}$ and hence show that $\tan 15^{\circ}+\cot 15^{\circ}=4$
Question 5
Evaluate $\tan \left\{(-1)^{n} \frac{\pi}{4}\right\}$, where $n$ is an integer.
Type 2
Question 6
If $\sin \alpha=\frac{1}{\sqrt{10}}, \sin \beta=\frac{1}{\sqrt{5}}$ (where $\alpha, \beta$ and $\alpha+\beta$ are positive acute angles). show that $\alpha+\beta=\frac{\pi}{4}$.
Page no 6.19
Question 7
If $\sin A=\frac{3}{5}$ and $\cos B=\frac{9}{41}, 0<A<\frac{\pi}{2}, 0<B<\frac{\pi}{2}$. Find the values of the following :
(i) $\sin (A+B)$
(ii) $\cos (A+B)$
(iii) $\sin (A-B)$
(iv) $\cos (A-B)$
Question 8
If $\sin A=\frac{1}{2}, \cos B=\frac{\sqrt{3}}{2}$, where $\frac{\pi}{2}<A<\pi$ and $0<B<\frac{\pi}{2}$, find the following :
(i) $\tan (A+B)$
(ii) $\tan (A-B)$
Question 9
If $\sin x=\frac{3}{5}, \cos y=-\frac{12}{13}$, where $\frac{\pi}{2}<x<\pi$ and $\frac{\pi}{2}<y<\pi$, show that $\sin (x+y)=-\frac{56}{65}$
Question 10
Evaluate :
(i) $\cos ^{2}\left(\frac{\pi}{4}+x\right)-\sin ^{2}\left(\frac{\pi}{4}-x\right)$
(ii) $\sin ^{2}\left(15^{\circ}+A\right)-\sin ^{2}\left(15^{\circ}-A\right)$
(iii) $\sin ^{2}\left(\frac{\pi}{8}+\frac{A}{2}\right)-\sin ^{2}\left(\frac{\pi}{8}-\frac{A}{2}\right)$
Question 11
Prove that
(i) $\cos \left(\frac{3 \pi}{2}+x\right) \cos (2 \pi+x)\left[\cot \left(\frac{3 \pi}{2}-x\right)+\cot (2 \pi+x)\right]=1$
(ii) $\frac{\cos (\pi+\theta) \cdot \cos (-\theta)}{\sin (\pi-\theta) \cdot \cos \left(\frac{\pi}{2}+\theta\right)}=\cot ^{2} \theta$
Question 12
Show that
(i) $\cos 70^{\circ}, \cos 10^{\circ}+\sin 70^{\circ} \cdot \sin 10^{\circ}=\frac{1}{2}$
(ii) $\sin 78^{\circ} \cdot \cos 18^{\circ}-\cos 78^{\circ} \cdot \sin 18^{\circ}=\frac{\sqrt{3}}{2}$
(iii) $\sin \left(40^{\circ}+\theta\right) \cdot \cos \left(10^{\circ}+\theta\right)-\cos \left(40^{\circ}+\theta\right) \cdot \sin \left(10^{\circ}+\theta\right)=\frac{1}{2}$
(iv) $\sin (n+1) x \sin (n+2) x+\cos (n+1) x \cos (n+2) x=\cos x$
(v) $\cos \left(\frac{\pi}{4}-x\right) \cos \left(\frac{\pi}{4}-y\right)-\sin \left(\frac{\pi}{4}-x\right) \sin \left(\frac{\pi}{4}-y\right)=\sin (x+y)$
(vi) $(\sin 3 x+\sin x) \sin x+(\cos 3 x-\cos x) \cos x=0$
(vii) $\frac{\tan \left(\frac{\pi}{4}+x\right)}{\tan \left(\frac{\pi}{4}-x\right)}=\left(\frac{1+\tan x}{1-\tan x}\right)^{2}$
Question 13
Prove that $\frac{\cos 20^{\circ}-\sin 20^{\circ}}{\cos 20^{\circ}+\sin 20^{\circ}}=\tan 25^{\circ}$.
Page no - 6.20
Question 14
Prove that $\cos 9^{\circ}+\cos 9^{\circ}=\sqrt{2} \sin 54^{\circ}$
Question 15
Prove that
$\frac{\tan A+\tan B}{\tan A-\tan B}=\frac{\sin (A+B)}{\sin (A-B)}$
Question 16
Prove that
(i) $\tan 8 \theta-\tan 6 \theta-\tan 2 \theta=\tan 8 \theta \tan 6 \theta \tan 2 \theta$
(ii) $\tan 9^{\circ}+\tan 36^{\circ}+\tan 9^{\circ} \tan 36^{\circ}=1$
(iii) $\tan 3 x-\tan 2 x-\tan x=\tan 3 x \tan 2 x \tan x$
Question 17
Show that
(i) $\frac{1}{\tan 3 A-\tan A}-\frac{1}{\cot 3 A-\cot A}=\cot 2 A$
(ii) $\frac{1}{\tan 3 A+\tan A}-\frac{1}{\cot 3 A+\cot A}=\cot 4 A$
Question 18
Prove that
(i) $\sin ^{2} 6 x-\sin ^{2} 4 x=\sin 2 x \sin 10 x$
(ii) $\tan (\alpha+\beta) \tan (\alpha-\beta)=\frac{\sin ^{2} \alpha-\sin ^{2} \beta}{\cos ^{2} \alpha-\sin ^{2} \beta}$
Question 19
Prove that :
(i) $\sin (A+B) \sin (A-B)+\sin (B+C) \sin (B-C)+\sin (C+A) \sin (C-A)=0$
(ii) $\tan \{(2 n+1) \pi+\theta\}+\tan \{(2 n+1) \pi-\theta \mid=0\}$
Type 3
Question 20
If $\tan \alpha=\frac{m}{m+1}, \tan \beta=\frac{1}{2 m+1}$, prove that $\alpha+\beta=\frac{\pi}{4}$
Question 21
If $A+B=\frac{\pi}{4}$, show that $(\cot A-1)(\cot B-1)=2$.
Question 22
If $8 \theta=\pi$, show that $\cos 7 \theta+\cot \theta=0$.
Question 23
If $\tan \alpha-\tan \beta=x$ and $\cot \beta-\cot \alpha=y$, prove that $\cot (\alpha-\beta)=\frac{x+y}{x y}$
Question 24
If $\tan \alpha=2 \tan \beta$, show that $\frac{\sin (\alpha+\beta)}{\sin (\alpha-\beta)}=3$.
Question 25
If $\cos A+\sin B=m$ and $\sin A+\cos B=n$. prove that $2 \sin (A+B)=m^{2}+n^{2}-2$.
Question 26
If $\tan x+\tan \left(\frac{\pi}{3}+x\right)+\tan \left(\frac{2 \pi}{3}+x\right)=3$, then prove that
Question 27
If a right angle be divided into three parts $\alpha, \beta$ and $\gamma$, prove that
$\cot \alpha=\frac{\tan \beta+\tan \gamma}{1-\tan \beta \tan \gamma} \text {. }$
Question 28
If $\sin \alpha, \sin \beta-\cos \alpha, \cos \beta=1$, show that $\tan \alpha+\tan \beta=0$.
Page no - 6.21
Question 29
If $\sin (\alpha+\beta)=1$ and $\sin (\alpha-\beta)=\frac{1}{2}$, where $0 \leq \alpha, \beta \leq \frac{\pi}{2}$, then find the values of $\tan (\alpha+2 \beta)$ and $\tan (2 \alpha+\beta)$
Question 30
If $m \tan \left(\theta-30^{\circ}\right)=n \tan \left(\theta+120^{\circ}\right)$. show that $\cos 2 \theta=\frac{m+n}{2(m-n)}$.
Question 31
If $\alpha+\beta=\theta$ and $\frac{\tan \alpha}{\tan \beta}=\frac{x}{y}$, prove that
$\sin (\alpha-\beta)=\frac{x-y}{x+y} \sin \theta$
Question 32
If $\alpha$ and $\beta$ are the solutions of the equation $a \cos \theta+b \sin \theta=c$, then show that
(i) $\cos (\alpha+\beta)=\frac{a^{2}-b^{2}}{a^{2}+b^{2}}$
(ii) $\cos (\alpha-\beta)=\frac{2 c^{2}-\left(a^{2}+b^{2}\right)}{a^{2}+b^{2}}$
Type 3
Question 33
Find the maximum and minimum values of the following expressions :
(i) $a \cos \theta-b \sin \theta$
(ii) $7 \cos \theta+24 \sin \theta$
Question 34
Show that $\sin 100^{\circ}-\sin 10^{\circ}$ is positive.
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