Question

Find the integrals of the functions $\frac{1-\cos x}{1+\cos x}$

Answer 1

$\frac{1-\cos x}{1+\cos x}=\frac{2{\sin }^2\frac{x}{2}}{2{\cos }^2\frac{x}{2}}$
$={\tan }^2\frac{x}{2}=({\sec }^2\frac{x}{2}-1)$
$\therefore \int \frac{1-\cos x}{1+\cos x}dx=\int ({\sec }^2\frac{x}{2}-1)dx$
$=[\frac{\tan \frac{x}{2}}{\frac{1}{2}}-x]+C$
$=2\tan \frac{x}{2}-x+C$