The value of ${\cot }^{-1}\left(\frac{\sqrt{1 + \sin \left(\mathrm{x}\right)} + \sqrt{1 - \sin \left(\mathrm{x}\right)}}{\sqrt{1 + \sin \left(\mathrm{x}\right)} - \sqrt{1 - \sin \left(\mathrm{x}\right)}}\right)$ is equal to

• $\frac{\mathrm{x}}{3}$

• $\frac{\mathrm{x}}{4}$

• 1

• $\frac{\mathrm{x}}{2}$

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\frac{1}{2}\right) = \frac{\mathrm{\pi }}{2}$, then value of x will be

• 0

• $\frac{1}{\sqrt{5}}$

• $\frac{2}{\sqrt{5}}$

• $\frac{\sqrt{3}}{2}$

The value of $\tan \left[2{\tan }^{-1}\left(\frac{1}{5}\right) - \frac{\mathrm{\pi }}{4}\right]$ is

• $\frac{17}{7}$

• $\frac{- 17}{7}$

• $\frac{7}{17}$

• $\frac{- 7}{17}$

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

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$

• None of these

165.

The value of $\sin \left(2{\tan }^{-1}\left(\frac{1}{3}\right)\right) + \cos \left({\tan }^{-1}\left(2\sqrt{2}\right)\right)$ will be

• $\frac{13}{12}$

• $\frac{14}{15}$

• $- \frac{14}{15}$

• $\frac{12}{13}$

${\tan }^{-1}\left(\frac{\mathrm{x} - 2}{\mathrm{x} - 3}\right) + {\tan }^{-1}\left(\frac{\mathrm{x} + 2}{\mathrm{x} + 3}\right) = \frac{\mathrm{\pi }}{4}$, then x = ?

• $\pm \frac{1}{\sqrt{2}}$

• $\pm \sqrt{\frac{7}{2}}$

• $\pm \sqrt{\frac{2}{7}}$

• None of these

${\sec }^2\left({\tan }^{ - 1}\left(2\right)\right) + {\mathrm{cosec}}^2 \left({\cot }^{ - 1}\left(3\right)\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• 3

• 10

• 15

• 20

${\sinh }^{ - 1}\left(\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^2}}\right) \mathrm{is} \mathrm{equal} \mathrm{to}$

• ${\coth }^{ - 1}\left(\mathrm{x}\right)$

• ${\sinh }^{ - 1}\left(\mathrm{x}\right)$

• $- {\tanh }^{ - 1}\left(\mathrm{x}\right)$

• ${\tanh }^{ - 1}\left(\mathrm{x}\right)$

$\mathrm{If} {\sin }^{-1}\left(\mathrm{x}\right) - {\cos }^{-1}\left(\mathrm{x}\right) = \frac{\mathrm{\pi }}{6}, \mathrm{then} \mathrm{x} \mathrm{is} \mathrm{equal} \mathrm{to}$

• $\frac{1}{2}$

• $\frac{\sqrt{3}}{2}$

• $- \frac{1}{2}$

• $- \frac{\sqrt{3}}{2}$

${\sinh }^{-1}\left(2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right)$ is equal to

• $\log \left(3 + \sqrt{8}\right)$

• $\log \left(3 - \sqrt{8}\right)$

• $\log \left(2 + \sqrt{18}\right)$

• $\log \left(\sqrt{8} + \sqrt{27}\right)$