JEE Math Practice Question

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

1. A.$$-\frac{1}{\sqrt{2}} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$$
2. B.$$\tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$$
3. C.$$\frac{1}{2} \tan ^{-1}\left(\frac{1}{\sqrt{2}}\right)$$
4. D.$$\frac{1}{\sqrt{2}} \tan ^{-1}\left(-\frac{1}{2}\right)$$