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Question
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?
Sum
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Solution
\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\} = \frac{- {cosec}^2 \left( \frac{\pi}{4} + \frac{x}{2} \right)}{2\cot\left( \frac{\pi}{4} + \frac{x}{2} \right)}\]
\[ = \frac{- 1}{2\cos\left( \frac{\pi}{4} + \frac{x}{2} \right)\sin\left( \frac{\pi}{4} + \frac{x}{2} \right)}\]
\[ = \frac{- 1}{\sin\left( \frac{\pi}{2} + x \right)}\]
\[ = \frac{- 1}{\cos x}\]
\[ = - \sec x\]
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Notes
The answer given at the back of the exercise in RD Sharma is incorrect.
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