$\sin (\cot ^{ - 1}x) = $

For the equation $\cos ^{-1} |x| + \cos ^{-1} |2x| = \pi$,the number of real solution$(s)$ is

$\tan \left( 90^o - \cot^{-1} \frac{1}{3} \right) = $

The value of $\sin (\cot ^{-1} x)$ is

Consider a function $f: [-1, 1] \to R$ where $f(x) = \alpha_1 \sin^{-1} x + \alpha_3 (\sin^{-1} x)^3 + \dots + \alpha_{2n+1} (\sin^{-1} x)^{2n+1} - \cot^{-1} x$,where $\alpha_i$ are positive constants and $n \in N < 100$. Then $f(x)$ is:

$\tan^{-1} \left[ \frac{\sqrt{2}}{\sqrt{3}} \cos \left( 5 \sin^{-1} \frac{1}{\sqrt{2}} \right) \right] = \dots$