I have had no experience so far with hyperbolic functions so any help will be appreciated. This is on the chapter of complex integration but I would especially appreciate it if you could turn this into a real integration problem. If not, one should just go with what he/she has.
Prove that: $\displaystyle \int_{-\infty}^{\infty} \frac{\cos(x)}{\cosh(x)} = \frac{\pi}{\cosh\left (\frac{\pi}{2} \right)}$