A) \[-\frac{1}{4}\]
B) \[\frac{1}{2}\]
C) \[-\frac{1}{2}\]
D) \[\frac{1}{4}\]
Correct Answer: A
Solution :
Let \[y={{\tan }^{-1}}\sqrt{\frac{1+\cos \frac{x}{2}}{1-\cos \frac{x}{2}}}={{\tan }^{-1}}\sqrt{\frac{2{{\cos }^{2}}\frac{x}{4}}{2{{\sin }^{2}}\frac{x}{4}}}\] \[y={{\tan }^{-1}}\cot \frac{x}{4}={{\tan }^{-1}}\tan \left( \frac{\pi }{2}-\frac{x}{4} \right)=\frac{\pi }{2}-\frac{x}{4}\] \ \[\frac{dy}{dx}=-\frac{1}{4}\].
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