If $\cos^{-1} \frac{3}{5} - \sin^{-1} \frac{4}{5} = \cos^{-1} x$,then $x = $

The trigonometric equation $\sin ^{-1} x = 2 \sin ^{-1} 2a$ has a real solution, if

If $\log_{\pi}x > 0$,then the value of $\log_{\pi}\left( \sin^{-1}\frac{2x}{1+x^2} + 2\tan^{-1}x \right)$ is equal to

The value of $\tan^{-1} \left( \frac{\sin 2 - 1}{\cos 2} \right)$ is equal to:

If $\operatorname{Tan}^{-1}\left[\frac{1}{1+1(2)}\right]+\operatorname{Tan}^{-1}\left[\frac{1}{1+(2)(3)}\right]+\operatorname{Tan}^{-1}\left[\frac{1}{1+(3)(4)}\right]+\cdots+\operatorname{Tan}^{-1}\left[\frac{1}{1+n(n+1)}\right]=\operatorname{Tan}^{-1} \theta$,then $\theta=$

If $\cos ^{-1} x - \cos ^{-1} \frac{y}{2} = \alpha$,where $-1 \leq x \leq 1, -2 \leq y \leq 2, x \leq \frac{y}{2}$,then for all $x, y$,$4x^2 - 4xy \cos \alpha + y^2$ is equal to