Show that

$\displaystyle \frac{\pi}{\cos(\pi z)} = 4\sum_{n=0}^{\infty} \frac{(-1)^n (2n+1)}{(2n+1)^2 - 4z^2}

$

So if you see my attempt here at first working around the tangent function decomposition [link], and making use of the identities $\displaystyle \cot(\pi z) + \tan(\frac{\pi}{2}z) = \frac{1}{\sin(z)}$, and $\displaystyle \cos(\pi z) = \sin(\pi(\frac{1}{2} - z))$, I haven't been going anywhere with this.

Appreciate any help. Thank you!