Question

If $y=x\tan \frac{x}{2},$ then $(1+\cos x)\frac{dy}{dx}-\sin x=$

• A. $xy$
• B. $y$
• C. $0$
• D. $x$

Answer 1

The correct option is D $x$

$y=x\tan \frac{x}{2}$
$\frac{dy}{dx}=x\frac{d}{dx}\left(\tan \frac{x}{2}\right)+\frac{d\left(x\right)}{dx}\cdot \tan \frac{x}{2}$
$=x\cdot \frac{1}{2}{\sec }^2\frac{x}{2}+\tan \frac{x}{2}$
$=\frac{x}{2{\cos }^2\frac{x}{2}}+\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}$
$=\frac{x+2\sin \frac{x}{2}\cos \frac{x}{2}}{2{\cos }^2\frac{x}{2}}$
$=\frac{x+\sin x}{1+\cos x}$
$\Rightarrow (1+\cos x)\frac{dy}{dx}-\sin x=x$