Basic Concepts in Inverse Trigonometric Functions

Suppose $f\left(x\right)=\tan \left({\sin }^{-1}\left(2x\right)\right)$. Find range of $f\left(x\right)$.

Find the value of ${\sin }^{-1}\left(\sin \frac{4\pi }{5}\right).$

The principal value of ${\tan }^{-1}\sqrt{3}-{\sec }^{-1}\left(-2\right)$ would be

Which of the following is the value of the given expression : ${\tan }^{-1}\left(1\right)+{\cos }^{-1}\left(-\frac{1}{2}\right)+{\sin }^{-1}\left(-\frac{1}{2}\right)$

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

If ${\cos }^{-1}\mathrm{x}+{\cos }^{-1}\mathrm{y}=\mathrm{\theta },$ show that ${\mathrm{x}}^2-2\mathrm{xycos\theta }+{\mathrm{y}}^2={\sin }^2\mathrm{\theta }$.

If ${\cos }^{-1}\frac{\mathrm{x}}{\mathrm{a}}+{\cos }^{-1}\frac{\mathrm{y}}{\mathrm{b}}=\mathrm{\theta }$, prove that $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{2\mathrm{xy}}{\mathrm{ab}}\mathrm{cos\theta }+\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}={\sin }^2\mathrm{\theta }$

Prove that:  ${\sin }^{-1}\sqrt{\frac{\mathrm{x}-\mathrm{b}}{\mathrm{a}-\mathrm{b}}}={\cos }^{-1}\sqrt{\frac{\mathrm{a}-\mathrm{x}}{\mathrm{a}-\mathrm{b}}}={\tan }^{-1}\sqrt{\frac{\mathrm{x}-\mathrm{b}}{\mathrm{a}-\mathrm{x}}}$

Prove that : $3{\tan }^{-1}\mathrm{x}={\tan }^{-1}\frac{3\mathrm{x}-{\mathrm{x}}^3}{1-3{\mathrm{x}}^2}$

Prove that : $3{\cos }^{-1}\mathrm{x}={\cos }^{-1}\left(4{\mathrm{x}}^3-3\mathrm{x}\right)$

Find the simplest value of $\tan \left[\frac{1}{2}{\sin }^{-1}\left(\frac{2\mathrm{x}}{1+{\mathrm{x}}^2}\right)+\frac{1}{2}{\cos }^{-1}\left(\frac{1-{\mathrm{y}}^2}{1+{\mathrm{y}}^2}\right)\right]$, where $\mathrm{xy}\ne 1.$

Solve: ${\cos }^{-1}\left({\mathrm{sincos}}^{-1}\mathrm{x}\right)=\frac{\mathrm{\pi }}{6}$

Solve: $\cos \left({\sin }^{-1}\mathrm{x}\right)=\frac{1}{9}$

Prove that: ${\tan }^{-1} \sqrt{\mathrm{x} } =\frac{1}{2} {\cos }^{-1} \frac{1-\mathrm{x}}{1+\mathrm{x}}$

Prove that, ${\sin }^{-1}\frac{1}{2}+2{\cot }^{-1}\frac{1}{\sqrt{3}}=\frac{5}{6}\mathrm{\pi }$

Prove that, $\sin \left[\frac{\mathrm{\pi }}{3}+{\sin }^{-1}\left(\frac{1}{2}\right)\right]=1$

Write down the general value of ${\cot }^{-1}\left(-\sqrt{3}\right)$

Write down the general value of ${\sec }^{-1}\left(-2\right)$