Question

Find the integral of the function
$\frac{1-\cos x}{1+\cos x}$

Answer 1

We have,

$I=\int \frac{1-\cos x}{1+\cos x}dx$ …… (1)

Since,

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

Therefore,

$I=\int \frac{1-(1-2{\sin }^2\frac{x}{2})}{1+2{\cos }^2\frac{x}{2}-1}dx$

$I=\int \frac{{\sin }^2\frac{x}{2}}{{\cos }^2\frac{x}{2}}dx$

$I=\int {\tan }^2\frac{x}{2}dx$

$I=\int ({\sec }^2\frac{x}{2}-1)dx$

$I=\frac{\tan \frac{x}{2}}{\frac{1}{2}}-x+C$

$I=2\tan \frac{x}{2}-x+C$

Hence, this is the answer.