## Inverse Trigonometric Functions Questions and Answers Part-3

1. The value of $\cos^{-1}x+\cos^{-1}\left(\frac{x}{2}+\frac{1}{2}\sqrt{3-3x^{2}}\right)\left(1/2\leq x\leq1\right)$
is equal to
a) $\pi/6$
b) $\pi/3$
c) $\pi$
d) 0

Explanation: Let cos–1 x = y. Then x = cos y, so that

2. $\sin^{-1}\left(\sin\left(\frac{2x^{2}+4}{1+x^{2}}\right)\right)<\pi-3$       if
a) $-1\leq x \leq 0$
b) $0\leq x \leq 1$
c) –1 < x < 1
d) x > 1

Explanation:

3. If $\tan^{-1}\frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}=\alpha$
then $x^{2}$ is equal to
a) $\sin\alpha$
b) $\cos2\alpha$
c) $\cos\alpha$
d) $\sin2\alpha$

Explanation: From the given relation we have

4. If $\tan^{-1}\frac{a}{x}+\tan^{-1}\frac{b}{x}+\tan^{-1}\frac{c}{x}+\tan^{-1}\frac{d}{x}=\frac{\pi}{2}$
then $x^{4}-x^{2}\sum ab+abcd$      is equal to
a) -1
b) 0
c) 1
d) 2

Explanation:

5. If $x_{1},x_{2},x_{3},x_{4}$   are roots of the equation $x^{4}-x^{3}\sin 2\beta+x^{2}\cos2\beta-x\cos\beta-\sin\beta=0$
then $\sum_{i=1}^{4}\tan^{-1}x_{i}$   is equal to
a) $\pi-\beta$
b) $\pi-2\beta$
c) $\left(\pi/2\right)-\beta$
d) $\left(\pi/2\right)-2\beta$

Explanation:

6. If $\alpha=3\sin^{-1}\left(\frac{6}{11}\right)$    and $\beta=3\cos^{-1}\left(\frac{4}{9}\right)$    where the inverse trigonometric functions take only the principal values, then the correct option (s) is (are)
a) $\cos\alpha < 0$
b) $\sin\beta < 0$
c) $\cos\left(\alpha+\beta\right) > 0$
d) All of the above

Explanation:

7. If the numerical value of $\tan \left(\cos^{-1}\left(4/5\right)+\tan^{-1}\left(2/3\right)\right)$      is a/b then
a) a + b = 23
b) a – b = 11
c) 3b = a + 1
d) All of the Above

Explanation:

8. If $a=\sin^{-1}\left(-\frac{\sqrt{2}}{2}\right)+\cos^{-1}\left(-\frac{1}{2}\right)$
and $b=\tan^{-1}\left(-\sqrt{3}\right)-\cot^{-1}\left(-\frac{1}{\sqrt{3}}\right)$
then
a) $a – b = 17\pi/12$
b) $a + b = 17\pi/12$
c) $a + b = -7\pi/12$
d) Both a and c

Explanation:

9. If $\alpha,\beta$ are the roots of the equation $\left(\tan^{-1}\left(x/5\right)\right)^{2}+\left(\sqrt{3}-1\right)\tan^{-1}\left(x/5\right)-\sqrt{3}=0 , \mid \alpha\mid >\mid\beta\mid$
then
a) $\alpha+\beta =-5\pi/12$
b) $\mid\alpha-\beta \mid=35\pi/12$
c) $\alpha\beta = -25\pi^{2}/12$
d) All of the Above

10. $\theta=\tan^{-1}\left(2\tan^{2}\theta\right)-\tan^{-1}\left(\left(1/3\right)\tan\theta\right)$
if tan $\theta$ is equal to