Relationship between cos and cosh

Theorem

The following formula holds: $$\cos(z)=\cosh(iz),$$ where $\cos$ is the cosine and $\cosh$ is the hyperbolic cosine.

Proof

From the definition of $\cosh$ and the definition of $\cos$, $$\cosh(iz)=\dfrac{e^{iz}+e^{-iz}}{2}=\cos(z),$$ as was to be shown.