$\cos (\tan ^{ - 1}x) = $

The principal value of $\sin ^{-1}\left(-\frac{1}{2}\right)$ is

Considering only the principal values of an inverse function,the set $A = \{x \geq 0 \mid \tan^{-1} x + \tan^{-1} 6x = \frac{\pi}{4}\}$

If $\theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$,then $\cos^{-1}(\sin \theta)$ is equal to

Find the principal value of $cosec^{-1}(2)$.

If $y = \cos^{-1}(\cos(|x| - f(x)))$,where $f(x) = \begin{cases} 1, & \text{if } x > 0 \\ -1, & \text{if } x < 0 \\ 0, & \text{if } x = 0 \end{cases}$,then $\left. \frac{dy}{dx} \right|_{x = \frac{5\pi}{4}}$ is