Domain of analyticity of a function

I must find the domain of analyticity of the function $\displaystyle f(z)=\frac{1}{\cosh z}$.

Attempt:

Since $\displaystyle \cosh (z)=\frac{e^{iz}+e^{-iz}}{2}$ is analytic over the whole complex plane, f is analytic on the whole complex plane except when $\displaystyle \frac{e^{iz}+e^{-iz}}{2}=0$.

This is equivalent to say that if $\displaystyle z=a+ib$, $\displaystyle a=\frac{\pi}{2}+\frac{k \pi}{2}$ with $\displaystyle k\in \mathbb{Z}$ and $\displaystyle \cos (2a)=-e^{b}$. I've searched in google the zeros of the complex function cosh and I didn't find anything (not even in wikipedia) that showed them. Of course likely I've missed something but still. I'm out of ideas about how to go further in the expression $\displaystyle \cos (2a)=-e^{b}$. What does this represent... Any idea?