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

• 1, - 1

• 1, 0

• 0, 1/2

• None of these

Range of f(x) = ${\sin }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{x}\right) + {\sec }^{-1}\left(\mathrm{x}\right)$ is

• $\left[\frac{\mathrm{\pi }}{4}, \frac{{\displaystyle 5\mathrm{\pi }}}{{\displaystyle 4}}\right]$

• $\left\{\frac{\mathrm{\pi }}{4}, \frac{3\mathrm{\pi }}{4}\right\}$

• $\left(\frac{\mathrm{\pi }}{4}, \frac{7\mathrm{\pi }}{4}\right)$

• None of these

If $\frac{1}{2}{\sin }^{-1}\left[\frac{3\sin \left(2\mathrm{\theta }\right)}{5 + 4\cos \left(2\mathrm{\theta }\right)}\right] = {\tan }^{-1}\left(x\right)$, then x is equal to

• 3$\cot \left(\mathrm{\theta }\right)$

• 3$\tan \left(\mathrm{\theta }\right)$

• $\tan \left(3\mathrm{\theta }\right)$

• $\frac{1}{3}\tan \left(\mathrm{\theta }\right)$

The value of $2{\tan }^{-1}\left(\csc \left({\tan }^{-1}\left(\mathrm{x}\right)\right) - \tan \left({\cot }^{-1}\left(\mathrm{x}\right)\right)\right)$ is

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

• $\tan \left(\mathrm{x}\right)$

• $\cot \left(\mathrm{x}\right)$

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

$2{\tan }^{-1}\left(\cos \left(\mathrm{x}\right)\right) = {\tan }^{-1}\left(2\csc \left(\mathrm{x}\right)\right)$, then value of x is

• $\frac{3\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• None of these

The principal value of ${\sin }^{-1}\left\{\sin \left(\frac{5\mathrm{\pi }}{6}\right)\right\}$ is

• $\frac{\mathrm{\pi }}{6}$

• $\frac{5\mathrm{\pi }}{6}$

•  $\frac{7\mathrm{\pi }}{6}$

• None of these

Range of the function y = ${\sin }^{-1}\left(\frac{{\mathrm{x}}^2}{1 + {\mathrm{x}}^2}\right)$, is

• $\left(0, \frac{\mathrm{\pi }}{2}\right)$

• $[0, \frac{\mathrm{\pi }}{2})$

• $(0, \frac{\mathrm{\pi }}{2}]$

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

${\cos }^{-1}\left(\frac{- 1}{2}\right) - 2{\sin }^{-1}\left(\frac{1}{2}\right) + 3{\cos }^{-1}\left(\frac{- 1}{\sqrt{2}}\right) - 4{\tan }^{-1}\left(- 1\right)$ equals

• $\frac{19\mathrm{\pi }}{12}$

• $\frac{35\mathrm{\pi }}{12}$

• $\frac{47\mathrm{\pi }}{12}$

• $\frac{43\mathrm{\pi }}{12}$

$\mathrm{x} = {\cos }^{-1}\left(\frac{1}{\sqrt{1 + {\mathrm{t}}^2}}\right), \mathrm{y} = {\sin }^{-1}\left(\frac{1}{\sqrt{1 + {\mathrm{t}}^2}}\right) \Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• 0

• tan(t)

• 1

• sin(t) cos(t)

The simplified expression of sin(tan^-1(x)) for any real number x is given by