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

If $\pi \le x \le 2\pi $,then ${\cos ^{ - 1}}(\cos x)$ is equal to

Statement-$1$: ${\cot ^{ - 1}}\left[ {\frac{{\log (e/{x^2})}}{{\log (ex^2)}}} \right] + {\cot ^{ - 1}}\left[ {\frac{{\log (ex^2)}}{{\log (e/{x^2})}}} \right] = \frac{\pi}{2}$
Statement-$2$: ${\tan ^{ - 1}}\left[ {\frac{{1 + \log {x^2}}}{{1 - \log {x^2}}}} \right] = {\tan ^{ - 1}}1 + {\tan ^{ - 1}}(\log {x^2})$

The value of $\sin \left(\cos ^{-1}\left(-\frac{1}{3}\right)-\sin ^{-1}\left(\frac{1}{3}\right)\right)$ is

The set of all values of $k$ for which $(\tan^{-1} x)^3 + (\cot^{-1} x)^3 = k \pi^3$,$x \in \mathbb{R}$,is the interval

The smallest and the largest values of ${\tan ^{ - 1}}\left( {\frac{{1 - x}}{{1 + x}}} \right)$ for $0 \le x \le 1$ are