A) \[\frac{1}{1+{{x}^{2}}}\]
B) \[\frac{-1}{1+{{x}^{2}}}\]
C) \[\frac{2}{1+{{x}^{2}}}\]
D) \[\]\[\frac{-2}{1+{{x}^{2}}}\]
Correct Answer: D
Solution :
Putting \[x=\cot \theta \] \[y={{\cos }^{-1}}\left( \frac{x-{{x}^{-1}}}{x+{{x}^{-1}}} \right)={{\cos }^{-1}}\left( \frac{{{x}^{2}}-1}{{{x}^{2}}+1} \right)\] \[={{\cos }^{-1}}(\cos 2\theta )=2\theta \Rightarrow \frac{dy}{dx}=\frac{-2}{1+{{x}^{2}}}\].
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