If $y=\operatorname{Sin}^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$ and $\frac{-3 \pi}{2} < x < \frac{-\pi}{2}$, then $\frac{d y}{d x}=$

• A

  $-\frac{\left|\operatorname{cosec} \frac{x}{2}\right|}{2 \sqrt{\sin ^2 \frac{x}{2}-\cos ^2 \frac{x}{2}}}$
• Correct answer
  B

  $\frac{\left|\sec \frac{x}{2}\right|}{2 \sqrt{\cos x}}$
• C

  $\frac{\left|\cos \frac{x}{2}\right|}{2 \sqrt{\cos x}}$
• D

  $\frac{\cos \frac{x}{2}}{\sqrt{\cos x}}$

Hence, the correct answer is option: 'B'.