# Cosine in terms of Hyperbolic Cosine

## Theorem

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

Then:

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

where:

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

## Proof

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

$\blacksquare$