The value of $\tan ^{-1}\left(\frac{1}{8}\right)+\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{5}\right)$ is

If $\tan^{-1} \left[ \frac{\sqrt{5} - 2\sqrt{6}}{1 + \sqrt{6}} \right] = \frac{\pi}{3} - \tan^{-1}(k)$, then $\sec^{-1}(k) = \dots$

If $y = \cot^{-1} \left( \frac{1 + x}{1 - x} \right)$,then $\frac{dy}{dx} = $

Suppose $S_a(x) = \operatorname{Sec}^{-1}\left(\frac{x}{a}\right) + \operatorname{Sec}^{-1}(a)$ for $a \neq 0$. If $S_a(x) = S_b(x)$ for $a \neq b$,then $x =$

Prove that $\tan ^{-1} x+\tan ^{-1} \frac{2 x}{1-x^{2}}=\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)$,where $|x| < \frac{1}{\sqrt{3}}$.

Considering only the principal values of the inverse trigonometric functions,the value of $\tan \left(\sin ^{-1}\left(\frac{3}{5}\right)-2 \cos ^{-1}\left(\frac{2}{\sqrt{5}}\right)\right)$ is