All the values of $x$ satisfying the equation $2 \tan^{-1} 2x = \sin^{-1} \left( \frac{4x}{1+4x^2} \right)$ lie in the interval

If $2 \tan^{-1}(\cos x) = \tan^{-1}(\csc^2 x)$,then $x =$

If $\angle A = 90^\circ$ in the triangle $ABC$,then $\tan^{-1}\left(\frac{c}{a+b}\right) + \tan^{-1}\left(\frac{b}{a+c}\right) = $

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:

If $\alpha > \beta > \gamma > 0$,then the expression $\cot ^{-1}\left\{\beta+\frac{(1+\beta^2)}{(\alpha-\beta)}\right\}+\cot ^{-1}\left\{\gamma+\frac{(1+\gamma^2)}{(\beta-\gamma)}\right\}+\cot ^{-1}\left\{\alpha+\frac{(1+\alpha^2)}{(\gamma-\alpha)}\right\}$ is equal to:

Find $\frac{dy}{dx}$ if $y = \sec^{-1}\left(\frac{1}{2x^2 - 1}\right)$,where $0 < x < \frac{1}{\sqrt{2}}$.