Properties 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

In a $\Delta ABC$, if $\angle A=\angle B=\frac{1}{2}\left({\sin }^{-1}\left(\frac{\sqrt{6}+1}{2\sqrt{3}}\right)+{\sin }^{-1}\left(\frac{1}{\sqrt{3}}\right)\right)$ and length of the side opposite to $\angle C$ is  $c=6·3^{\frac{1}{4}}$, then the area of $\Delta ABC$ is

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

If ${\sin }^{-1}\left(\mathrm{x}-\frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{x}}^3}{4}-\dots \dots .\right)+{\cos }^{-1}\left({\mathrm{x}}^2-\frac{{\mathrm{x}}^4}{2}+\frac{{\mathrm{x}}^6}{4}-\dots \dots .\right)=\frac{\mathrm{\pi }}{2}$ for $0<\left|\mathrm{x}\right|<\sqrt{2}$ then $\mathrm{x}$ equals to

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

For $x\in (-1,1]$, the number of solutions of the equation ${\sin }^{-1}x=2{\tan }^{-1}x$ is equal to

If ${\sin }^{-1}x=2{\tan }^{-1}x$, then number of integral values of $x$ is equal to:

The number of solution(s) of the equation ${\tan }^{-1}\left(x+\frac{2}{x}\right)-{\tan }^{-1}\left(\frac{4}{x}\right)-{\tan }^{-1}\left(x-\frac{2}{x}\right)=0$ is