Verify that each equation is an identity

[kuahji]

The question is "Verify that each equation is an identity."

Problem
tan 8k - tan 8k tan^2 4k = 2 tan 4k

The first thing I tried was to factor out the tangents.

tan 8k (1 - tan^2 4k)

then if I'm doing this correctly

tan 8k (1 - tan 2k) (1 + tan 2k)

But from here I'm stuck, that is if I'm on the right track.

 

[neutrino]

"tan 8k (1 - tan^2 4k)"

From here use the double-angle formula.

 

[sutupidmath]

tan 8k (1 - tan 4k) (1 + tan 4k), this is ok,although you did not need to factorize (1-tan^24k)at all.
now notice that we can write down tan8k=tan2*4k=2tan4k/(1-tan^2 4k), after substitution look what can be canceled out.
I am assuming that you already know why
tan2*4k=2tan4k/(1-tan^2 4k), is so, right??

Edit: like neutrino said, i just elaborated it a little bit more

 

[neutrino]

tan 8k (1 - tan 2k) (1 + tan 2k), this is ok

It is not.

 

[sutupidmath]

It is not.

Yeah, sorry by mistake, because i just copied this from the op and did not acyally look at it.!
What i meant was (1-tan4k)(1+tan4k).

sorry!

 

[HallsofIvy]

It might also be a good idea not to lose track of the right side of the equation!