Question

How do you integrate `(1-tanx)/(1+tanx) dx`?

Answer 1

Rewrite as sines and cosines.

Explanation:

`(1-tan x)/(1+tan x)=(1-(sin x)/(cos x ))/(1+(sin x)/(cos x)) = (cos x - sin x )/(cos x + sin x ) = (cos x cos (pi/4)-sin x sin (pi/4))/(cos x cos (pi/4) + sin x sin (pi/4) )=cot(x+pi/4)`
Note that `sin(pi/4)=cos(pi/4)=1/sqrt 2`