Simplify $\tan ^{-1}\left[\frac{a \cos x-b \sin x}{b \cos x+a \sin x}\right],$ if $\frac{a}{b} \tan x > -1$.

The value of $\tan ^{-1} \frac{1}{3}+\tan ^{-1} \frac{1}{5}+\tan ^{-1} \frac{1}{7}+\tan ^{-1} \frac{1}{8}$ is $ . . . . . . $

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

Prove $\cos ^{-1} \frac{4}{5} + \cos ^{-1} \frac{12}{13} = \cos ^{-1} \frac{33}{65}$

If $4 \sin ^{-1} x + \cos ^{-1} x = \pi$,then $x = $

Let $(a, b) \subset (0, 2\pi)$ be the largest interval for which $\sin^{-1}(\sin \theta) - \cos^{-1}(\sin \theta) > 0$ holds for $\theta \in (0, 2\pi)$. If $\alpha x^2 + \beta x + \sin^{-1}(x^2 - 6x + 10) + \cos^{-1}(x^2 - 6x + 10) = 0$ and $\alpha - \beta = b - a$,then $\alpha$ is equal to: