Inverse Trigonometric Functions Questions and Answers Part-2

1. If $a\sin^{-1}x-b\cos^{-1}x=c$     , then $a\sin^{-1}x+b\cos^{-1}x$      is equal to
a) 0
b) $\frac{\pi ab+c\left(b-a\right)}{a+b}$
c) $\pi/2$
d) $\frac{\pi ab+c\left(a-b\right)}{a+b}$

Explanation:

2. If $\frac{1}{2}\sin^{-1}\left[\frac{3\sin2\theta}{5+4\cos 2\theta}\right]=\tan^{-1}x$
then x =
a) $\tan3\theta$
b) $3\tan\theta$
c) $\left(1/3\right)\tan\theta$
d) $3 \cot\theta$

Explanation:

3. If x > 0, y > 0 and x > y, then $\tan^{-1}\left(x/y\right)+\tan^{-1}[\left(x+y\right)/\left(x-y\right)]$
is equal to
a) $-\pi/4$
b) $\pi/4$
c) $3\pi/4$
d) none of these

Explanation:

4. The equation $2 \cos^{-1}x=\sin^{-1}\left(2x\sqrt{1-x^{2}}\right)$       is valid for all values of x satisfying
a) $-1\leq x\leq 1$
b) $0\leq x\leq 1$
c) $-0\leq x\leq 1/\sqrt{2}$
d) $1/\sqrt{2}\leq x\leq 1$

Explanation: If we denote cos- 1x by y, then

5. If $\tan^{-1}\frac{1}{1+2}+\tan^{-1}\frac{1}{1+\left(2\right)\left(3\right)}+\tan^{-1}\frac{1}{1+\left(3\right)\left(4\right)}+....+\tan^{-1}\frac{1}{1+n\left(n+1\right)}=\tan^{-1}\theta$
then $\theta=$
a) $\frac{n}{n+1}$
b) $\frac{n+1}{n+2}$
c) $\frac{n}{n+2}$
d) $\frac{n-1}{n+2}$

Explanation:

6. A root of the equation $17x^{2}+17x \tan\left[2\tan^{-1}\left(1/5\right)-\pi/4\right]-10=0$
is
a) 10/17
b) -1
c) -7/17
d) 1

Explanation:

7. The value of $\sin\left(2\tan^{-1}\left(1/3\right)\right)+\cos\left(\tan^{-1}2\sqrt{2}\right)$
is
a) 12/13
b) 13/14
c) 14/15
d) none of these

Explanation:

8. The sum of the infinite series $\cot^{-1}2+\cot^{-1}8+\cot^{-1}18+\cot^{-1}32+....$
is equal to
a) $\pi/3$
b) $\pi/4$
c) $\pi/6$
d) $\pi/8$

Explanation: The nth term of the series can be written as

9. The value of $\sin^{-1}\left\{\cot\left(\sin^{-1}\sqrt{\frac{2-\sqrt{3}}{4}}+\cos^{-1}\frac{\sqrt{12}}{4}+\sec^{-1}\sqrt{2}\right)\right\}$
is equal to
a) $\pi/4$
b) $\pi/6$
c) 0
d) $\pi/2$

10. If $u=\cot^{-1}\sqrt{\cos\alpha}-\tan^{-1}\sqrt{\cos\alpha},$
a) $\tan^{2}\left(\alpha/2\right)$
b) $\cot^{2}\left(\alpha/2\right)$
c) $\tan^{2}\alpha$
d) $\cot^{2}\alpha$