Question

Differentiate ${\tan }^{-1}\sqrt{\frac{1+\cos x}{1-\cos x}}$

• A. $\frac{-1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{8}$
• D. None of these

Answer 1

The correct option is A $\frac{-1}{2}$

Let $y={\tan }^{-1}\sqrt{\frac{1+\cos x}{1-\cos x}}$

$={\tan }^{-1}\sqrt{\frac{2{\cos }^2\frac{x}{2}}{2{\sin }^2\frac{x}{2}}}$

$={\tan }^{-1}\cot \left(\frac{x}{2}\right)$

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

$=\frac{\pi }{2}-\frac{x}{2}$

$\Rightarrow \frac{dy}{dx}=-\frac{1}{2}$