subtracting trig

Question

I can't seem to get passed this hurdle, and I just literally have no idea where to start from.

$$2\cos^2(2x) - 2\sin^2(2x) = \text{ ?}$$

I'm being told it's $2\cos(4x)$, but I have no clue how to they got there. If someone could give me a head start (not the answer) on what trig identity to use, I'd be really grateful. Thanks

Answer 1

$$\cos2A=\cos^{2}A-\sin^{2}A$$ To say any more would be writing down the answer.

Answer 2

The equation you want to make use of is:

$$\cos(2x) = \cos^2x-\sin^2x.$$

This can be derived from the relation

$$\cos(x\pm y) = \cos x\cos y \mp \sin x\sin y$$

by setting $y = x$.