Understanding the Quadruple Angle Identity for Cosine

[physicsgeek54]

Homework Statement

Simplify: cos(4θ)

Homework Equations

cos(2θ)=2cos^2(θ)-1
sin(2θ)=2sinθcosθ

The Attempt at a Solution

First, I broke it into cos(2θ+2θ). Then I expanded it and got cos(2θ)cos(2θ)-sin(2θ)sin(2θ). I then expanded that and got (4cos^4(θ)-4cos^2(θ)+1)-(4sin^2(θ)+4sin(θ)cos(θ)+cos^2). I was supposed to get 8cos^4(θ)-8cos^2(θ)+1. I see that I'm off but I don't know where I went wrong.

 

[jedishrfu]

from your work:

the term (4sin^2(θ)+4sin(θ)cos(θ)+cos^2) isn't right for sin^2(2x) = (2sin(x)cos(x))^2 = 4 sin^2(x)cos^2(x)

a simpler derivation would be:

cos(4x) = 2 cos^2(2x) - 1 right?

next we look at the cos(2x) factor: cos(2x) = 2cos^2(x) -1

and plug back into the 2cos^2(2x) - 1

 

[physicsgeek54]

Oh, thanks for catching my mistake. I think I can figure it out now but I don't see how you got from cos(4x) to 2 cos^2(2x)-1.