Help Understanding Trig Identity

[lntz]

Hello there,

I have a problem I'm hoping someone can help me with. I'm writing a bit of code for computing the value of pi that converges faster than a previous piece that relies on the leibniz series.

Anyway, I'm struggling with showing how this identity arises. tan(2t) = 2 * tan(t) / 1 - tan²(t)

So far I've got to this point;

sin(2θ) = 2sin(θ)cos(θ) and,
cos(2θ) = cos²(θ) - sin²(θ)

tan(2θ) = sin(2θ) / cos(2θ)

= 2sin(θ) cos(θ) / cos²(θ) - sin²(θ)

From there, I know that this is the step I'm supposed to take but I'm struggling to make sense of it. =[ 2sin(θ)cos(θ) / cos²(θ) ] * 1 / 1 - tan²(θ

Doing that step brings me back to the identity, but why it works is what I don't understand. I have been told that it is dividing by cos²θ but I must be doing something wrong because it doesn't seem to work out for me. when dividing through by cos²θ I get 2sin(θ)cos(θ) / 1 - tan²θ

I would really appreciate someone explaining this without making any large jumps in the reasoning. (I shouldn't be doing maths at this time, but it's really bugging me)

Thanks!

 

[SteamKing]

Your algebra is OK to this step:

tan(2θ) = sin(2θ) / cos(2θ)

= 2sin(θ) cos(θ) / (cos^2(θ) - sin^2(θ))

Now, what you need to do is divide both the numerator and the denominator by cos^2(theta) and then simplify.