## Trigonometry Questions and Answers Part-9

1. $\left(\sin\alpha+ cosec\alpha\right)^{2}+\left(\cos\alpha+\sec\alpha\right)^{2}-\left(\tan^{2}\alpha+\cot ^{2}\alpha\right)$
for all values of $\alpha$ is equal to
a) 0
b) 2
c) 4
d) 7

Explanation:

2. The expression $3\left[\sin^{4}\left(\frac{3\pi}{2}-\alpha\right)+\sin^{4}\left(3\pi+\alpha\right)\right]-2\left[\sin^{6}\left(\frac{\pi}{2}+\alpha\right)+\sin^{6}\left(5\pi-\alpha\right)\right]$
is equal to
a) 0
b) -1
c) 1
d) 3

Explanation:

3. If $\sin\left(x+3\alpha\right)=3\sin\left(\alpha-x\right)$      , then
a) $\tan x = \tan\alpha$
b) $\tan x = \tan^{2}\alpha$
c) $\tan x = \tan^{3}\alpha$
d) $\tan x =3 \tan\alpha$

Explanation:

4. If $\frac{\cos \left(\theta-\alpha\right)}{\sin\left(\theta+\alpha\right)}=\frac{m+1}{m-1}$
then m is equal to
a) $\tan\left(\frac{\pi}{4}-\theta\right)\tan\left(\frac{\pi}{4}-\alpha\right)$
b) $\tan\left(\frac{\pi}{4}-\theta\right)\tan\left(\frac{\pi}{4}+\alpha\right)$
c) $\tan\left(\frac{\pi}{4}+\theta\right)\tan\left(\frac{\pi}{4}+\alpha\right)$
d) $\tan\left(\frac{\pi}{4}+\theta\right)\tan\left(\frac{\pi}{4}-\alpha\right)$

Explanation:

5. The value of tan 130° tan 140° is equal to
a) -1
b) 1
c) $1/\sqrt{3}$
d) $1+\sqrt{3}$

Explanation:

6. If cos 1° cos 2° cos 3° ..... cos 179° = x + 1, then x is equal to
a) -1
b) 0
c) 1
d) none of these

Explanation:

7. If $-\pi/4\leq x<\pi/4$    and $\frac{1+\tan x}{1-\tan x} =1+\sin 2x$
then tan x is equal to
a) -1
b) $-1/\sqrt{3}$
c) 1
d) 2

Explanation:

8. If $\tan\alpha =\frac{m}{m-1}$    and $\tan\beta =\frac{1}{2m-1}$    then
a) $\alpha+\beta =\pi/4$
b) $\alpha-\beta =\pi/4$
c) $\alpha+\beta =\pi/6$
d) $\alpha-\beta =\pi/3$

Explanation:

9. If $m \tan\left(\theta-30^{\circ}\right)=n \tan\left(\theta+120^{\circ}\right)$
then $\frac{m-n}{m+n}$    is equal to
a) $2 \cos 2\theta$
b) $2 \sin^{2}\theta$
c) $1/\left(2 \cos 2\theta\right)$
d) $1/\left(2 \sin 2\theta\right)$

10. If $\cos x+\sin x=\sqrt{2} \cos x$     , then $\tan ^{2}x+2\tan x$