Proof of a Zeta function identity

You don't need to use complex analysis since

$$\log (\sin (x))=\log (x)-\sum _{n=1}^{\infty } \frac{ \zeta (2 n)\,x^{2 n}}{n \,\pi ^{2 n}}\tag{1}$$

can be integrated between $0$ and $\pi$, which in turn can be derived by integrating the expansion of $\cot x$ below:

$$ \cot (x)=\frac{1}{x}-2 x \sum _{n=1}^{\infty } \frac{1}{(n\,\pi)^2-x^2}$$

See for example: G. Boros and V. H. Moll, Irresistible Integrals, Cambridge: Cambridge University Press, 2004, page 10.

(1) can also be derived using the sine product formula.