Question

Find the integration of (1-cos2x)/(1+cos2x)

Answer 1

Given:

The expression is:

`(1-\cos 2x)/(1+\cos 2x)`

To find:

The integration of the given expression.

Solution:

We need to find the integration of
`(1-\cos 2x)/(1+\cos 2x)`.

Let us consider,

`I=\int (1-\cos 2x)/(1+\cos 2x)dx`

`I=\int (2\sin^2x)/(2\cos^2x)dx`
`[\because 1+\cos 2x=2\cos^2x,1-\cos 2x=2\sin^2x]`

`I=\int (\sin^2x)/(\cos^2x)dx`

`I=\int \tan^2xdx`
`\left[\because \tan \theta =(\sin \theta)/(\cos \theta)\right]`

It can be written as:

`I=\int (\sec^2x-1)dx`
`[\because 1+\tan^2 \theta =\sec^2 \theta]`

`I=\int \sec^2xdx-\int 1dx`

`I=\tan x-x+C`

Therefore, the integration of
`(1-\cos 2x)/(1+\cos 2x)` is
`I=\tan x-x+C`.