Question

$\frac{d}{dx}\left({\tan }^{-1}\left(\sqrt{\frac{1+\cos \left({\displaystyle \frac{x}{2}}\right)}{1-\cos \left({\displaystyle \frac{x}{2}}\right)}}\right)\right)$ is equal to

• A. $-\frac{1}{4}$
• B. $\frac{1}{4}$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Answer 1

The correct option is A

$-\frac{1}{4}$

Explanation for correct option:

Step-1: Simplify the given data.

Given, $\frac{d}{dx}\left({\tan }^{-1}\left(\sqrt{\frac{1+\cos \left({\displaystyle \frac{x}{2}}\right)}{1-\cos \left({\displaystyle \frac{x}{2}}\right)}}\right)\right)$

$$\begin{gathered} =\frac{d}{dx}\left({\tan }^{-1}\left(\sqrt{\frac{2{\cos }^2\left({\displaystyle \frac{x}{4}}\right)}{2{\sin }^2\left({\displaystyle \frac{x}{4}}\right)}}\right)\right) \\ =\frac{d}{dx}\left({\tan }^{-1}\left(\cot \left(\frac{x}{4}\right)\right)\right) \\ =\frac{d}{dx}\left({\tan }^{-1}\left(\tan \left(\frac{\pi }{2}-\frac{x}{4}\right)\right)\right) \\ =\frac{d}{dx}\left(\frac{\pi }{2}-\frac{x}{4}\right) \end{gathered}$$

Step-2: Differentiate w.r.t $x$.

$$\begin{gathered} =0-\frac{1}{4}\times \left(1\right) \\ =-\frac{1}{4} \end{gathered}$$

Hence, correct answer is option $\left(A\right)$