# Thread: ? Each time you see (cos(2x))^2, replace it with 1/2 + 1/2cos(4x) ?

1. ## ? Each time you see (cos(2x))^2, replace it with 1/2 + 1/2cos(4x) ?

I understand sin(2x)^2 = 1/2 - 1/2cos(2x), due to the double number identity: cos2x = 1 - 2sin(x)^2.

***[My question is how does (cos(2x))^2 = 1/2 + 1/2cos(4x) ???]***

I've tried reworking the identity every which way and can't seem to find out how the two are equivalent. Thanks in advance!

2. Originally Posted by estex198
I understand sin(2x)^2 = 1/2 - 1/2cos(2x), due to the double number identity: cos2x = 1 - 2sin(x)^2.

***[My question is how does (cos(2x))^2 = 1/2 + 1/2cos(4x) ???]***

I've tried reworking the identity every which way and can't seem to find out how the two are equivalent. Thanks in advance!
Every which way ....? What about this way:

$\cos (2A) = \cos^2 A - \sin^2 A = 2 \cos^2 A - 1 \Rightarrow \cos^2 A = \frac{1}{2} (\cos (2A) + 1)$.

$A = 2x$ in your case.

3. mr fantastic, you made perfect sense of it. Thanks a million. Now I just need to figure out how to get my hair back in my head. Thanks again!

- Rusty