Question

If \( y = \operatorname{Sin}^{-1} \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \) and \( -\frac{3\pi}{2}<x<-\frac{\pi}{2} \), then \( \frac{dy}{dx} \) is:

• A. \( -\frac{\sqrt{\operatorname{cosec}\frac{x}{2}}}{2\sqrt{\sin^2\frac{x}{2}-\cos^2\frac{x}{2}}} \)
• B. \( \frac{|\sec\frac{x}{2}|}{2\sqrt{\cos x}} \)
• C. \( \frac{|\cos\frac{x}{2}|}{2\sqrt{\cos x}} \)
• D. \( \frac{\cos x}{2\sqrt{\cos x}} \)

Hint:

For inverse trigonometric derivatives, express the argument in terms of fundamental trigonometric identities before differentiating.

Answer 1

The Correct Option is B

Rewriting the given function: \[ y = \operatorname{Sin}^{-1} \left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right) \] Using trigonometric simplifications, \[ y = \operatorname{Sin}^{-1}(\sec(x/2)) \] Differentiating: \[ \frac{dy}{dx} = \frac{|\sec(x/2)|}{2\sqrt{\cos x}} \] Thus, the correct answer is: \[ \frac{|\sec\frac{x}{2}|}{2\sqrt{\cos x}} \]