$\frac{d}{dx}[\tan^{-1}(\cot x) + \cot^{-1}(\tan x)] = $

The value of $\cot^{-1} \left[ \frac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}} \right]$, where $x \in (0, \frac{\pi}{2})$ is...

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 \sin ^{-1} x + 3 \cos ^{-1} x = \frac{2 \pi}{5}$ is:

If $\tan ^{-1}\left(\frac{x-1}{x-2}\right)+\tan ^{-1}\left(\frac{x+1}{x+2}\right)=\frac{\pi}{4}$,then the values of $x$ are

If $\sum_{n=1}^k \tan ^{-1}\left(\frac{1}{n^2+3 n+3}\right)=\tan ^{-1} \alpha$, then $\alpha=$

The sum of the first $n$ terms of the series $\cot^{-1} 3 + \cot^{-1} 7 + \cot^{-1} 13 + \cot^{-1} 21 + \dots$ is given by: