$\frac{d}{d x}\left[\cos ^2\left(\cot ^{-1} \sqrt{\frac{2+x}{2-x}}\right)\right]$ is

• A

  $-\frac{3}{4}$
• B

  $-\frac{1}{2}$
• C

  $\frac{1}{2}$
• Correct answer
  D

  $\frac{1}{4}$

Solution

$\because \frac{d}{d x}\left[\cos ^2\left(\cot ^{-1} \sqrt{\frac{2+x}{2-x}}\right)\right]$ $=\frac{d}{d x}\left(\cos \left(\cos ^{-1}\left(\frac{\sqrt{2+x}}{2}\right)\right)\right)^2$ $=\frac{d}{d x}\left(\frac{2+x}{4}\right)=\frac{1}{4}$

Hence, the correct answer is option: 'D'.