## Trigonometry Questions and Answers Part-10

1. If $A=\sin ^{8}\theta+\cos ^{14}\theta$    , then for all values of $\theta$
a) A > 1
b) $A \geq 1$
c) A < 1
d) $A \leq 1$

Explanation:

Note : A > 0 as A = 0 if sin $\theta$ = cos $\theta$ = 0 which is not possible.

2. If $\pi/2< \alpha <\pi$   , then the expression $\sqrt{\frac{1-\sin\alpha}{1+\sin\alpha}}+\sqrt{\frac{1+\sin\alpha}{1-\sin\alpha}}$
is equal to
a) $2/\cos\alpha$
b) $-2/\cos\alpha$
c) $2/\sin\alpha$
d) $2\tan\alpha$

Explanation:

3. If $0<\theta <\pi/2$   and $\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+cosec\theta$
is equal to 7, then sin 2$\theta$  is a root of the equation
a) $x^{2}+44x+36=0$
b) $x^{2}-44x-36=0$
c) $x^{2}-44x+36=0$
d) $x^{2}+44x-36=0$

Explanation:

4. If $\frac{\cos\left(\theta_{1}-\theta_{2}\right)}{\cos\left(\theta_{1}+\theta_{2}\right)}+\frac{\cos\left(\theta_{3}+\theta_{4}\right)}{\cos\left(\theta_{3}-\theta_{4}\right)}=0$
then $\tan\theta_{1}\tan\theta_{2}\tan\theta_{3} \tan\theta_{4}$      is equal to
a) -1
b) 1
c) 2
d) 4

Explanation:

5. If $\cos 2\beta=\frac{\cos\left(\alpha+\gamma\right)}{\cos\left(\alpha-\gamma\right)}$
then $\tan\alpha,\tan\beta$     and $\tan\gamma$  are in
a) A.P
b) G.P
c) H.P
d) none of these

Explanation:

6. If $x\cos \alpha+y \sin\alpha =x\cos\beta+y\sin\beta=2a\left(0<\alpha,\beta<\pi/2\right)$
then
a) $\cos \alpha+\cos\beta =\frac{4ax}{x^{2}+y^{2}}$
b) $\cos \alpha\cos\beta =\frac{4a^{2}-y^{2}}{x^{2}+y^{2}}$
c) $\sin \alpha+\sin\beta =\frac{4ay}{x^{2}+y^{2}}$
d) All of the Above

Explanation:

7. If $y=\frac{\sqrt{1-\sin 4A}+1}{\sqrt{1+\sin4A}-1}$
then one of the values of y is,
a) $-\tan A$
b) $\cot A$
c) $\tan \left(\frac{\pi}{4}+A\right)$
d) All of the Above

Explanation:

8. For $0<\theta<\pi/2,\tan\theta+\tan2\theta+\tan 3\theta =0$         if
a) $\tan\theta=0$
b) $\tan\theta\tan2\theta=2$
c) $\tan3\theta=0$
d) Both b and c

Explanation:

9. If $\sin\theta\left(1+\sin\theta\right)+\cos\theta\left(1+\cos\theta\right)=x$
and $\sin\theta\left(1-\sin\theta\right)+\cos\theta\left(1-\cos\theta\right)=y$
then
a) $x^{2}-2x=\sin2\theta$
b) $y^{2}+2y=\sin2\theta$
c) $xy=\sin2\theta$
d) All of the Above

10. If $\tan 6\theta=\frac{p}{q}$    , then
a) $p\sin6\theta+q\cos6\theta=\sqrt{p^{2}+q^{2}}$
b) $\frac{1}{2}$ (p cosec $2\theta-q\sec 2\theta)=\sqrt{p^{2}+q^{2}}$
c) $p\cos6\theta+q\sin6\theta=\frac{2pq}{\sqrt{p^{2}+q^{2}}}$