Question

If $y={\tan }^{-1}(\sec x-\tan x),x\in (0,\frac{\pi }{2}),$ then $\frac{dy}{dx}$ is equal to

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

Answer 1

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

$y={\tan }^{-1}(\sec x-\tan x)$
$\because \sec x-\tan x=\tan (\frac{\pi }{4}-\frac{x}{2})$
$\therefore y={\tan }^{-1}\left(\tan (\frac{\pi }{4}-\frac{x}{2})\right)$
$\because \frac{\pi }{4}-\frac{x}{2}\in (0,\frac{\pi }{4})$
$\therefore y=\frac{\pi }{4}-\frac{x}{2}$
$\frac{dy}{dx}=-\frac{1}{2}$