# Hyperbolic Cosine in terms of Cosine

## Theorem

Let $z \in \C$ be a complex number.

Then:

$\cosh z = \map \cos {i z}$

where:

$\cos$ denotes the complex cosine
$\cosh$ denotes the hyperbolic cosine
$i$ is the imaginary unit: $i^2 = -1$.

## Proof

 $\displaystyle \map \cos {i z}$ $=$ $\displaystyle \frac {e^{i \paren {i z} } + e^{-i \paren {i z} } } 2$ Cosine Exponential Formulation $\displaystyle$ $=$ $\displaystyle \frac {e^{-z} + e^z} 2$ $i^2 = -1$ $\displaystyle$ $=$ $\displaystyle \cosh z$ Definition of Hyperbolic Cosine

$\blacksquare$