Evaluate: $\sec^{-1} x + \operatorname{cosec}^{-1} x + \cos^{-1}(x^{-1}) + \sin^{-1}(x^{-1})$ (where $|x| > 1, x \in R$)

If the sum of all the solutions of $\tan ^{-1}\left(\frac{2 x}{1-x^2}\right)+\cot ^{-1}\left(\frac{1-x^2}{2 x}\right)=\frac{\pi}{3}$ for $-1 < x < 1$ and $x \neq 0$ is $\alpha-\frac{4}{\sqrt{3}}$,then $\alpha$ is equal to $..........$.

The value of $\cot \left( {\sum\limits_{n = 1}^{19} {{{\cot }^{ - 1}}\left( {1 + \sum\limits_{p = 1}^n {2p} } \right)} } \right)$ is

If $y = \tan^{-1} \sqrt{\frac{1 + \cos x}{1 - \cos x}}$,then $\frac{dy}{dx}$ is

If ${\cot ^{ - 1}}\frac{n}{\pi } > \frac{\pi }{6},\,\,n \in N$,then the maximum value of $n$ is

If $\sin ^{-1} x < \cos ^{-1} x$,then