$\sec^{-1}[\sec(-30^o)] = $ ....... $^o$

If $y = \sin^{-1}\left(\frac{2^{x+1}}{1+4^x}\right)$ and $\frac{dy}{dx} = \frac{2^{x+1} \log 2}{f(x)}$,then $f(0) = $ . . . . . .

The value of $\cos^{-1}(\frac{1}{2}) + 2\sin^{-1}(\frac{1}{2})$ is equal to

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:

${\left[ {\sin \left( {{{\tan }^{ - 1}}\frac{3}{4}} \right)} \right]^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: