For $x \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, if $y(x)=\int \frac{\operatorname{cosec} x+\sin x}{\operatorname{cosec} x \sec x+\tan x \sin ^2 x} d x$, and $\lim _\limits{x \rightarrow\left(\frac{\pi}{2}\right)^{-}} y(x)=0$ then $y\left(\frac{\pi}{4}\right)$ is equal to

Question

$-\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

$\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

$\frac{1}{2} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$

$\frac{1}{\sqrt{2}} \tan ^{-1}\left(-\frac{1}{2}\right)$

Solution

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