Show that
$\displaystyle \sum_{n=1}^\infty \frac{\zeta(2n)}{n 2^{2n}} = \ln \left( \frac{\pi}{2}\right)$
I used the equality : $\displaystyle \Gamma(s) \zeta(s) = \int^{\infty}_0 \frac{t^{s-1}}{e^t-1}\,dt$
and got $\displaystyle \,2\int^{\infty}_0 \frac{\cosh(\frac{t}{2})-1}{t(e^t-1)}\, dt$
I am still trying to find this integral ,