If $- \frac{\mathrm{\pi }}{2} < {\sin }^{-1}\left(\mathrm{x}\right) < \frac{\mathrm{\pi }}{2}$, then $\tan \left({\sin }^{-1}\left(\mathrm{x}\right)\right)$ is equal to

• $\frac{\mathrm{x}}{1 - {\mathrm{x}}^2}$

• $\frac{\mathrm{x}}{1 +\hspace{0.17em}{\mathrm{x}}^2}$

• $\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}}$

• $\frac{1}{\sqrt{1 - {\mathrm{x}}^2}}$

If $\mathrm{y} = {\tan }^{-1}\left(\mathrm{x}\right) + {\sec }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\mathrm{x}\right) + {\csc }^{-1}\left(\mathrm{x}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{{\mathrm{x}}^2 - 1}{{\mathrm{x}}^2 + 1}$

• $\mathrm{\pi }$

• 0

• 1

The doamin of ${\sin }^{-1}\left[{\log }_2\left(\frac{\mathrm{x}}{12}\right)\right]$ is

• [2, 12]

• [- 1, 1]

• $\left[\frac{1}{3}, 24\right]$

• [6, 24]

If ${\tan }^{-1}\left(\mathrm{x} + 2\right) + {\tan }^{-1}\left(\mathrm{x} - 2\right) - {\tan }^{-1}\left(\frac{1}{2}\right) = 0$, then one of the values of x is equal to

• - 1

• 5

• $\frac{1}{2}$

• 1

175.

The value of ${\sin }^{-1}\left(\frac{4}{5}\right) + 2{\tan }^{-1}\left(\frac{1}{3}\right)$ is equal to

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

The value of ${\sin }^{-1}\left(\cos \left(4095°\right)\right)$ is equal to

• $- \frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $- \frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

If $2{\sin }^{-1}\left(\mathrm{x}\right) - {\cos }^{-1}\left(\mathrm{x}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to

• $\frac{1}{\sqrt{2}}$

• $- \frac{1}{\sqrt{2}}$

• $\frac{- \sqrt{3}}{2}$

• $\frac{\sqrt{3}}{2}$

The value of ${\tan }^{-1}\left(2\right) + {\tan }^{-1}\left(3\right)$ is equal to

• $\frac{3\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• ${\tan }^{-1}\left(6\right)$

The value of ${\tan }^{-1}\left(\frac{\sqrt{3}}{2}\right) + {\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$ is equal to

• ${\tan }^{-1}\left(\frac{5}{\sqrt{3}}\right)$

• ${\tan }^{-1}\left(\frac{2}{\sqrt{3}}\right)$

• ${\tan }^{-1}\left(\frac{1}{2}\right)$

• ${\tan }^{-1}\left(\frac{1}{3\sqrt{3}}\right)$

The value of $\cos \left({\sin }^{-1}\left(\frac{2}{3}\right)\right)$ is equal to

• $\frac{\sqrt{3}}{5}$

• $\frac{5}{3}$

• $\frac{5}{\sqrt{3}}$

• $\frac{\sqrt{5}}{3}$