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. $-\frac{1}{2}$
• E. $-\frac{x}{2}$

Answer 1

Answer: (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,
$\frac{dy}{dx}=-\frac{1}{2}$