The number of solutions of $\sin ^{-1} x+\sin ^{-1}(1-x)=\cos ^{-1} x$ is

The value of $\cos ^{-1} \left[ \cot \left( \sin ^{-1} \sqrt{\frac{2-\sqrt{3}}{4}} \right) + \cos ^{-1} \left( \frac{\sqrt{12}}{4} \right) + \sec ^{-1} \sqrt{2} \right]$ is

If $\alpha$ and $\beta$ (where $\alpha > \beta$) are two zeroes of the equation $3\cos^{-1}\left(x^2 - 5x - \frac{11}{2}\right) = \pi$,then $(\alpha^2 + \beta^3)$ is -

The value of $\sec ^2(\tan ^{-1} 2)+\operatorname{cosec}^2(\cot ^{-1} 3)$ is

If the equation $\sin^{-1}(x - 1) + \cos^{-1}(x - 3) + \tan^{-1}\left(\frac{x}{-x^2 + 2}\right) = m$ holds,then the value of $m$ is

If $\theta = 2 \tan^{-1} \frac{1}{8} + 2 \tan^{-1} \frac{1}{5} + \tan^{-1} \frac{1}{7}$ and $\tan \frac{\theta}{2} = \sqrt{m} + \sqrt{n}$,where $m$ and $n$ are positive integers such that $m < n$,then $(m^n + n^m)^{m+n}$ is equal to