$\tan \left(2 \tan^{-1}\left(\frac{1}{3}\right)+\tan^{-1}\left(\frac{1}{7}\right)\right) = $

$\sum\limits_{m = 1}^n {{{\tan }^{ - 1}}} \left( {\frac{{2m}}{{{m^4} + {m^2} + 2}}} \right)$ is equal to

The value of $x$ such that $\sin \left(2 \tan ^{-1} \frac{3}{4}\right)=\cos \left(2 \tan ^{-1} x\right)$ is

If $x$ takes a negative permissible value,then $\sin^{-1} x$ is equal to

If $\cos^{-1} x = \alpha$ $(0 < x < 1)$ and $\sin^{-1} (2 x \sqrt{1 - x^2}) + \sec^{-1} (\frac{1}{2 x^2 - 1}) = \frac{2 \pi}{3}$,then $\alpha$ is

$S = \tan^{-1}\left( \frac{1}{n^2 + n + 1} \right) + \tan^{-1}\left( \frac{1}{n^2 + 3n + 3} \right) + \dots + \tan^{-1}\left( \frac{1}{1 + (n + 19)(n + 20)} \right)$,then $\tan S$ is equal to