Identities of Inverse Trigonometric Functions

If $y={\tan }^{-1}\frac{1}{x^2+x+1}+{\tan }^{-1}\frac{1}{x^2+3x+3}+{\tan }^{-1}\frac{1}{x^2+5x+7}+{\tan }^{-1}\frac{1}{x^2+7x+13}+\dots \dots \dots \dots$ up to $n$ terms, then $\frac{dy}{dx}$ is

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

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

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

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

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

If ${\tan }^{-1}x-{\tan }^{-1}y={\tan }^{-1}\left(\frac{x-y}{1+xy}\right)$ is true, then value of $xy$ is

The solution of ${\tan }^{-1}\left(2x\right)+{\tan }^{-1}\left(3x\right)=\frac{\pi }{4}$ is

If $-\frac{\mathrm{\pi }}{2}<{\sin }^{-1}x<\frac{\mathrm{\pi }}{2}$, then $\tan \left({\sin }^{-1}x\right)$ is equal to

If ${\cos }^{-1}x={\sin }^{-1}\left(3x\right)$, then $x=$

If $\theta =2{\tan }^{-1}\frac{1}{8}+2{\tan }^{-1}\frac{1}{5}+{\tan }^{-1}\frac{1}{7}$ and $\tan \frac{\theta }{2}=-\sqrt{m}+\sqrt{n}$ where $m$ and $n$ are positive integers such that $m<n$ then ${\left(m^n+n^m\right)}^{m+n}=$

$\tan \left[\frac{1}{2}{\sin }^{-1}\left(\frac{4}{5}\right)+\frac{1}{2}{\cos }^{-1}\left(\frac{15}{17}\right)\right]$ is equal to

If $a$ and $b$ are positive integers satisfying ${\cot }^{-1}a+{\cot }^{-1}b={\cot }^{-1}7$, then the number of ordered pairs of $\left(a, b\right)$ is

If two angles of $∆ABC$ are ${\tan }^{-1}\frac{1}{3}$ and ${\tan }^{-1}\frac{1}{2}$, then the third angle of $∆ABC$ is

$\cos \left({\cot }^{-1}\left(cosec\left({\cos }^{-1}a\right)\right)\right)=\_\_\_\_\_$

If $y=\sum _{r=1}^n{\tan }^{-1}\left(\frac{1}{x^2+(2r-1)x+r(r-1)+1}\right)$, then the value of $\frac{dy}{dx}$ at $x=0$ is 

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