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

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{2\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

Solve for $\mathrm{x} {\tan }^{-1}\left(\frac{1 - \mathrm{x}}{1 + \mathrm{x}}\right) = \frac{1}{2}{\tan }^{-1}\left(\mathrm{x}\right) . \mathrm{x} > 0$

• - 1

• $\sqrt{3}$

• $\frac{1}{\sqrt{3}}$

• 1

If $\frac{{\left(\mathrm{x} + 1\right)}^2}{{\mathrm{x}}^3 + \mathrm{x}} = \frac{\mathrm{A}}{\mathrm{x}} + \frac{\mathrm{Bx} + \mathrm{C}}{{\mathrm{x}}^2 + 1}$, then ${\csc }^{-1}\left(\frac{1}{\mathrm{A}}\right) + {\cot }^{-1}\left(\frac{1}{\mathrm{B}}\right) + {\sec }^{-1}\left(\mathrm{C}\right)$ is equal to

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

• 0

144.

If $\mathrm{\alpha } \le 2{\sin }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{x}\right) \le \mathrm{\beta }$, then

• $\mathrm{\alpha } = 0, \mathrm{\beta } = \mathrm{\pi }$

• $\mathrm{\alpha } = \frac{- \mathrm{\pi }}{2}, \mathrm{\beta } = \frac{\mathrm{\pi }}{2}$

• $\mathrm{\alpha } = 0, \mathrm{\beta } = 2\mathrm{\pi }$

• $\mathrm{\alpha } = \frac{- \mathrm{\pi }}{2}, \mathrm{\beta } = \frac{3\mathrm{\pi }}{2}$

The value of ${\sin }^{-1}\left(\cos \left(\frac{53\mathrm{\pi }}{5}\right)\right)$ is

• $\frac{3\mathrm{\pi }}{5}$

• $\frac{- 3\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{10}$

• $\frac{- \mathrm{\pi }}{10}$

If $3{\tan }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\mathrm{x}\right) = \mathrm{\pi }$, then x is equal to

• 0

• 1

• - 1

• 1/2

The simplified form of ${\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) - {\tan }^{-1}\left(\frac{\mathrm{x} - \mathrm{y}}{\mathrm{x} + \mathrm{y}}\right)$ is equal to

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\mathrm{\pi }$

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(\mathrm{y}\right) = \frac{\mathrm{\pi }}{2}$, then x² is equal to

• 1 - y²

• y²

• 0

• $\sqrt{1 - \mathrm{y}}$

The range of ${\sec }^{-1}\left(\mathrm{x}\right)$ is

• $\left[0, \mathrm{\pi }\right] - \left\{\frac{\mathrm{\pi }}{2}\right\}$

• $\left[- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right]$

• $\left\{- \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right\}$

• $\left[0, \mathrm{\pi }\right]$

If ${\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) = \frac{4\mathrm{\pi }}{5}$, then ${\cot }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\mathrm{y}\right)$ is equal to

• $\frac{2\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{5}$

• $\frac{3\mathrm{\pi }}{5}$

• $\mathrm{\pi }$