Question

Using `t=tan(theta/2)`, prove that `tan(x/2+pi/4)+tan(x/2-pi/4)=2tanx`?

Answer 1

For the proof we use
`tan(x/2+pi/4)=(1+tan(x/2))/(1-tan(x/2))`
and
`tan(x/2-pi/4)=(tan(x/2)-1)/(tan(x/2)+1)`
`tan(pi/4)=1`

Explanation:

So we get

`((tan(x/2)+1)^2+(tan(x/2)-1)*(1-tan(x/2)))/(1-tan^2(x/2))`
This is
`(tan^2(x/2)+1+2tan(x/2)+tan(x/2)-1-tan^2(x/2)+tan(x/2))/(1-tan^2(x/2))`
and this is

`(4tan(x/2))/(1-tan^2(x/2))=2tan(x)`