$\sin \left[ \frac{\pi }{2} - \sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) \right] = $

If $y = \cot^{-1} \left[ \frac{\sqrt{1 + \sin x} + \sqrt{1 - \sin x}}{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}} \right]$,then $\frac{dy}{dx} = $

If $\theta = \tan^{-1}\left(\frac{1}{3}\right) + \tan^{-1}\left(\frac{1}{7}\right) + \tan^{-1}\left(\frac{1}{13}\right) + \tan^{-1}\left(\frac{1}{21}\right) + \tan^{-1}\left(\frac{1}{31}\right)$,then $\tan \theta =$

If $x_1, x_2, x_3$ are the real roots of the equation $x^3-x^2 \tan \theta+x \tan ^2 \theta+\tan \theta=0$ and $0 < \theta < \frac{\pi}{4}$,then the value of $\tan ^{-1} x_1+\tan ^{-1} x_2+\tan ^{-1} x_3$ at $\theta=\frac{\pi}{12}$ is

If for $x \in \left(0, \frac{1}{4}\right)$,the derivative of $\tan^{-1}\left(\frac{6x\sqrt{x}}{1-9x^3}\right)$ is $\sqrt{x} \cdot g(x)$,then $g(x)$ equals

If $\theta = \sin^{-1}x + \cos^{-1}x - \tan^{-1}x$ for $x \ge 0$,then the smallest interval in which $\theta$ lies is given by