Question

If $y=\log \left[\tan (\frac{\pi }{4}+\frac{x}{2})\right]$ then $\frac{dy}{dx}=$

• A. $\sec x$
• B. $\sin x$
• C. $cosecx$
• D. $\sec \frac{x}{2}$

Answer 1

The correct option is A $\sec x$

$y=\log \left[\tan (\frac{\pi }{4}+\frac{x}{2})\right]$

$\therefore \frac{dy}{dx}=\frac{1}{\tan (\frac{\pi }{4}+\frac{x}{2})}{\sec }^2(\frac{\pi }{4}+\frac{x}{2})\cdot \frac{1}{2}$
$=\frac{1}{2\sin (\frac{\pi }{4}+\frac{x}{2})\cos (\frac{\pi }{4}+\frac{x}{2})}=\frac{1}{\sin \left[2(\frac{\pi }{4}+\frac{x}{2})\right]}=\frac{1}{\cos x}=\sec x$