A) \[\frac{1}{2}\]
B) \[\frac{2}{3}\]
C) 3
D) 1
Correct Answer: A
Solution :
\[y={{\cot }^{-1}}\left[ \frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \right]\] \[={{\cot }^{-1}}\left[ \frac{2+2\cos x}{2\sin x} \right]={{\cot }^{-1}}\left[ \frac{1+\cos x}{\sin x} \right]\] \[={{\cot }^{-1}}\left[ \cot \frac{x}{2} \right]=\frac{x}{2}\] \[\therefore \frac{dy}{dx}=\frac{1}{2}\] .
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