The value of $\cos ^{-1}\left(\tan \left(\frac{7 \pi}{4}\right)\right)$ is

Consider the following statements:
Statement-$I$: $\operatorname{Cosh}^{-1} x = \operatorname{Tanh}^{-1} x$ has no solution.
Statement-$II$: $\operatorname{Cosh}^{-1} x = \operatorname{Coth}^{-1} x$ has only one solution.
The correct answer is:

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

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:

If $x = \tan^{-1} \left(\frac{1}{5}\right) + \tan^{-1} \left(\frac{1}{8}\right)$, then $\frac{\sin x + \cos x}{\tan x} = $

If the area of the region bounded by the parabola $y^2 = 4kx$ and the line $x = k$, (where $k > 0$) is $\frac{128}{3}$ sq. units, then the value of $\sin^{-1} \left( \frac{2}{k} \right)$ is equal to...