A friend asked me to try to help, and I can't get it.

I think the identity you are trying to solve for is:

\cos^2(x) = \frac{1}{2} + \frac{1}{2} \cos(2x)

Remember the identities:
\cos^2(x) + \sin^2(x) = 1
and
\cos (2x) = \cos^2(x) - \sin^2(x)