$\cot \left[ \cos^{-1} \left( \frac{7}{25} \right) \right] = $

If $\frac{(x + 1)^2}{x^3 + x} = \frac{A}{x} + \frac{Bx + C}{x^2 + 1}$,then $\sin^{-1}\left(\frac{A}{C}\right) = $

The principal value of $\cot^{-1}\left(\frac{-1}{\sqrt{3}}\right)$ is equal to . . . . . . .

Solve $\tan ^{-1} 2 x+\tan ^{-1} 3 x=\frac{\pi}{4}$

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(\tan \frac{5 \pi}{6}\right)+\cos ^{-1}\left(\cos \frac{13 \pi}{6}\right) = $