# Secant in terms of Hyperbolic Secant

## Theorem

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

Then:

$\sec z = \map \sech {i z}$

where:

$\sec$ denotes the secant function
$\sech$ denotes the hyperbolic secant
$i$ is the imaginary unit: $i^2 = -1$.

## Proof

 $\displaystyle \sec z$ $=$ $\displaystyle \frac 1 {\cos z}$ Definition of Complex Secant Function $\displaystyle$ $=$ $\displaystyle \frac 1 {\map \cosh {i z} }$ Cosine in terms of Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \map \sech {i z}$ Definition of Hyperbolic Secant

$\blacksquare$