Question

How do you prove the following trig identity: `tan(x + 45°) - tan(45° - x) ≡ 2tan2x`?

Answer 1

Use `tan(a+b) = (tana+tanb)/(1-tanatanb)`

and `tan(a-b) = (tana-tanb)/(1+tanatanb)`

and `tan (2a) = tan(a+a) = (2tana)/(1-tan^2a)`

And also use: `tan (pi/4) =1`.

Other than that it's just algebra.

Once you get to

`(tanx + 1)/(1-tanx)-(1-tanx)/(1+tanx)`

you'll want a common denominator, so you get

`((tanx+1)^2-(1-tanx)^2)/(1-tan^2x)`

Now do the algebra to get:

`(4tanx)/(1-tan^2x) = 2( (2tanx)/(1-tan^2x)) = 2 tan2x`