Question

If $ y={{\cot }^{-1}}\left( \tan \frac{x}{2} \right), $ then $ \frac{dy}{dx} $ is equal to

• A. $ \frac{1}{2} $
• B. $ 0 $
• C. $ \frac{x}{2} $
• D. $ -\frac{1}{2} $

Answer 1

The Correct Option is D

$ y={{\cot }^{-1}}\left( \tan \frac{x}{2} \right) $ $ y={{\cot }^{-1}}\cot \left( \frac{\pi }{2}-\frac{x}{2} \right) $ $ y=\frac{\pi }{2}-\frac{x}{2} $
Differentiating w.r.t. $ x $ on both sides,
$ \frac{dy}{dx}=-\frac{1}{2} $