Prove that $$ \ln(\cos x) = \sum_{n=1}^{\infty} \frac{(-1)^n \cos(2n x)}{n} - \ln 2. $$
I've tried to expand $ \cos(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} $ and do the expansion on $\ln(1 + u) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} u^n}{n}$.
