Properties of Inverse Trigonometric Functions

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

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

For $x\in \left(0,\frac{\mathrm{\pi }}{2}\right)$, the expression ${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\sin x-\frac{1}{\sqrt{2}}\cos x\right)=$

Find ${\tan }^{-1}\left(4\right)-{\tan }^{-1}\left(1\right)$.

Find ${\tan }^{-1}\left(3\right)-{\tan }^{-1}\left(1\right)$.

Find ${\tan }^{-1}\left(2\right)-{\tan }^{-1}\left(1\right)$.

Find ${\tan }^{-1}\left(3\right)-{\tan }^{-1}\left(2\right)$.

Find the value of: $\tan \left(2{\tan }^{-1}\frac{1}{5}-\frac{\pi }{4}\right)$

Number of integral solutions of equation $\left|{\cos }^{-1}\cos \left|x\right|\right|=\tan {\tan }^{-1}x$ is/are 

If $\left|x\right|<1$, then $\sin \left({\tan }^{-1}x\right)$ is equal to 

The value of ${\cos }^{-1}\left(\cos \frac{\pi }{7}\cos \frac{\pi }{17}-\sin \frac{\pi }{7}\sin \frac{\pi }{17}\right)$ is

If ${\sin }^{-1}x+{\sin }^{-1}\frac{1}{x}+{\cos }^{-1}x+{\cos }^{-1}\frac{1}{x}=k\pi$, then $k$ is

If ${\tan }^{-1}\left(\tan \frac{2\pi }{3}\right)=k°$, then $k=$

Prove that : $2{\tan }^{-1}\left[\tan \left(45°-\alpha \right)\tan \frac{\beta }{2}\right]={\cos }^{-1}\left(\frac{\sin 2\alpha +\cos \beta }{1+\sin 2\alpha \cos \beta }\right)$

Prove that: ${\sin }^{-1}\frac{3}{5}-{\cos }^{-1}\frac{12}{13}={\sin }^{-1}\frac{16}{65}$

Solve the equation ${\cos }^{-1}x+{\cos }^{-1}2x=\frac{2\pi }{3}$.

Solve for $x$, If ${\sin }^{-1}x+{\sin }^{-1}2x=\frac{\pi }{3}$