The value of $\tan^{-1} \left( \frac{\cos(\frac{19\pi}{4}) - 1}{\sin(\frac{\pi}{4})} \right)$ is equal to

If $\cos^{-1} x = y$, then $\dots \dots \dots$

If the range of $\operatorname{sech}^{-1} x + \operatorname{cosech}^{-1} x$ is $[a, b]$, then

The principal value of $\tan^{-1} \left( \cot \frac{43\pi}{4} \right)$ is

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

If $2 \tan^{-1}(\cos x) = \tan^{-1}(2 \operatorname{cosec} x)$,then the value of $x$ is