If ${\cos }^{-1}\left(\frac{\mathrm{x}}{\mathrm{a}}\right) + {\cos }^{-1}\left(\frac{\mathrm{y}}{\mathrm{b}}\right) = \mathrm{\alpha }\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{2\mathrm{xy}}{\mathrm{ab}}\cos \left(\mathrm{\alpha }\right) + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}$, then $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2} - \frac{2\mathrm{xy}}{\mathrm{ab}}\cos \left(\mathrm{\alpha }\right) + \frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}$ is equal to

• ${\sin }^2\left(\mathrm{\alpha }\right)$

• ${\mathrm{acos}}^2\left(\mathrm{\alpha }\right)$

• ${\mathrm{atan}}^2\left(\mathrm{\alpha }\right)$

• ${\mathrm{acot}}^2\left(\mathrm{\alpha }\right)$

The domain of the function ${\sin }^{-1}\left({\log }_2\left(\frac{{\mathrm{x}}^2}{2}\right)\right)$ is

• $\left[- 2, 2\right] ~ \left(- 1, 1\right)$

• $\left[- 1, 2\right] ~ \left\{0\right\}$

• [1,2]

• $\left[- 2, 2\right] ~ \left\{0\right\}$

The domain of ${\sin }^{-1}\left[{\log }_3\left(\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)\right]$ is

• [1, 9]

• [- 1, 9]

• [- 9, 1]

• [- 9, - 1]

The trigonometric equation ${\sin }^{-1}\left(\mathrm{x}\right) = 2{\sin }^{-1}\left(\mathrm{a}\right)$, has a solution for

• $\frac{1}{2} < \left|\mathrm{a}\right| \le \frac{1}{\sqrt{2}}$

• all real values of a

• $\left|\mathrm{a}\right| \le \frac{1}{\sqrt{2}}$

• $\left|\mathrm{a}\right| \ge \frac{1}{\sqrt{2}}$

155.

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

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

Domain of y = ${\cot }^{-1}\left(\mathrm{x}\right)$ is

• $\left(- \infty , 0\right)$

• $\left(0, \infty \right)$

• $\left(- \infty , \infty \right)$

• None of these

$\sin \left[{\cot }^{-1}\left\{{\tan }^{-1}\left(\mathrm{x}\right)\right\}\right]$ is equal to

• 1

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

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

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

If $\sin \left({\sin }^{-1}\left(\frac{1}{5}\right) + {\cos }^{-1}\left(\mathrm{x}\right)\right) = 1$, then x is

• $\frac{1}{5}$

• $\frac{2}{5}$

• $\frac{3}{5}$

• $\frac{\mathrm{\pi }}{2}$

The value of $\tan \left[{\cos }^{-1}\left(\frac{4}{5}\right) + {\tan }^{-1}\left(\frac{2}{3}\right)\right]$ will be

• $\frac{6}{11}$

• $\frac{6}{17}$

• $\frac{11}{6}$

• $\frac{17}{6}$

The value of ${\sin }^{-1}\left(\frac{3}{5}\right) + {\sin }^{-1}\left(\frac{8}{17}\right)$ is

• ${\cos }^{-1}\left(\frac{36}{85}\right)$

• ${\sin }^{-1}\left(\frac{15}{85}\right)$

• ${\sin }^{-1}\left(\frac{36}{85}\right)$

• ${\cos }^{-1}\left(\frac{15}{85}\right)$