The solution of $\sin^{-1} x - \sin^{-1} 2x = \pm \frac{\pi}{3}$ is

If $2 \tan^{-1}(\cos x) = \tan^{-1}(2 \operatorname{cosec} x)$,then the value of $x$ is

If $0 < x < 1$,then $\sqrt{1+x^2} [\{x \cos (\cot ^{-1} x)+\sin (\cot ^{-1} x)\}^2-1]^{\frac{1}{2}}$ is equal to

If $y = \cos^2 [\cot^{-1} (\sqrt{\frac{1-x}{1+x}})]$, then $\frac{dy}{dx} = \dots$

Let $0 < \alpha < 1$, $\beta = \frac{1}{3\alpha}$, and $\tan^{-1}(1 - \alpha) + \tan^{-1}(1 - \beta) = \frac{\pi}{4}$. Then $6(\alpha + \beta)$ is equal to:

The sum to infinite terms of the series $\tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{2}{9}\right) + \dots + \tan^{-1}\left(\frac{2^{n-1}}{1+2^{2n-1}}\right) + \dots$ is