Properties of Inverse Trigonometric Functions

Let $y={\sin }^{-1}\left(\sin 8\right)-{\tan }^{-1}\left(\tan 10\right)+{\cos }^{-1}\left(\cos 12\right)$$-{\sec }^{-1}\left(\sec 9\right)+{\cot }^{-1}\left(\cot 6\right)-{\mathrm{cosec}}^{-1}\left(\mathrm{cosec}7\right)$. If $y$ simplifies to $a\pi +b$, then $\left(a-b\right)$ is

Evaluate:   ${\tan }^{-1}\left(\frac{1}{2}\right)+{\tan }^1\left(\frac{1}{5}\right)+{\tan }^1\left(\frac{1}{8}\right)$

The principal value of ${\cos }^{-1}\left(\cos \frac{7\pi }{6}\right)$ is:

${\tan }^{-1}\frac{x-1}{x-2}+{\tan }^{-1}\frac{x+1}{x+2}=\frac{\pi }{4},$ then the value of $x$ could be

$\cos \left({\sin }^{-1}\frac{3}{5}+{\cot }^{-1}\frac{3}{2}\right)$  would be equal to:

Which of the following is the value of given expression : $\frac{9\pi }{8}-\frac{9}{4}{\sin }^{-1}\left(\frac{1}{3}\right)$

The solution of the equation ${\tan }^{-1}2x+{\tan }^{-1}3x=\frac{\pi }{4}$ would be:

The value of $x$ for which  $\sin \left[{\cot }^{-1}\left(1+x\right)\right]=\cos \left({\tan }^{-1}x\right)$

The number of real solutions of ${\tan }^{-1}\sqrt{x\left(x+1\right)}+{\sin }^{-1}\sqrt{x^2+x+1}=\frac{\pi }{2}$  is –

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

Solve: ${\sin }^{-1}\frac{2\mathrm{a}}{1+{\mathrm{a}}^2}+{\sin }^{-1}\frac{2\mathrm{b}}{1+{\mathrm{b}}^2}={\cos }^{-1}\frac{1-{\mathrm{x}}^2}{1+{\mathrm{x}}^2}$

Prove that : $\tan \frac{1}{2} \left( {\tan }^{-1} \mathrm{x} + \tan { }^{-1} \frac{1}{\mathrm{x}}\right)=1$

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

Prove that, ${\sin }^{-1}\frac{1}{\sqrt{5}}+{\tan }^{-1}\frac{1}{3}=\frac{\mathrm{\pi }}{4}$

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

Prove that, $\frac{1}{2}{\sin }^{-1}\frac{2a}{1+a^2}+\frac{1}{2}{\cos }^{-1}\frac{1-b^2}{1+b^2}={\tan }^{-1}\frac{a+b}{1-ab}$