Question

Differentiate : \[\tan^{- 1} \left( \frac{1 + \cos x}{\sin x} \right)\] with respect to x .

Answer 1

\[\text {Let y  }= \tan^{- 1} \left( \frac{1 + \cos x}{\sin x} \right) . \text { Then }, \]

\[ \Rightarrow y = \tan^{- 1} \left( \frac{2 \cos^2 \frac{x}{2}}{2\sin\frac{x}{2} . \cos\frac{x}{2}} \right)\]

\[ \Rightarrow y = \tan^{- 1} \left( \cot\frac{x}{2} \right)\]

\[ \Rightarrow y = \tan^{- 1} \left\{ \tan\left( \frac{\pi}{2} - \frac{x}{2} \right) \right\}\]

\[ \Rightarrow y = \frac{\pi}{2} - \frac{x}{2}\]

\[ \therefore \frac{dy}{dx} = 0 - \frac{1}{2} = - \frac{1}{2}\]