Question

Differentiate:

$\frac{{(1-\cos x)}^2}{{\sin }^{2}x}$

Answer 1

Let $y=\frac{{(1-\cos x)}^2}{{\sin }^{2}x}$

$y={\left(\frac{2{\sin }^2\frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}}\right)}^2$

$y={\tan }^2\frac{x}{2}$

Differentiate on both sides w.r.t. x

$\frac{dy}{dx}=2\tan \frac{x}{2}.{\sec }^2\frac{x}{2}.\frac{1}{2}$

$\therefore \frac{dy}{dx}=\tan \frac{x}{2}.{\sec }^2\frac{x}{2}$