Question

\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\}\] ?

Answer 1

\[\log\left\{ \cot\left( \frac{\pi}{4} + \frac{x}{2} \right) \right\} = \frac{- {cosec}^2 \left( \frac{\pi}{4} + \frac{x}{2} \right)}{2\cot\left( \frac{\pi}{4} + \frac{x}{2} \right)}\]

\[ = \frac{- 1}{2\cos\left( \frac{\pi}{4} + \frac{x}{2} \right)\sin\left( \frac{\pi}{4} + \frac{x}{2} \right)}\]

\[ = \frac{- 1}{\sin\left( \frac{\pi}{2} + x \right)}\]

\[ = \frac{- 1}{\cos x}\]

\[ = - \sec x\]