# Cosecant Exponential Formulation

## Theorem

Let $z$ be a complex number.

Let $\csc z$ denote the cosecant function and $i$ denote the imaginary unit: $i^2 = -1$.

Then:

$\csc z = \dfrac {2 i} {e^{i z} - e^{-i z} }$

## Proof

 $\displaystyle \csc z$ $=$ $\displaystyle \frac 1 {\sin z}$ $\quad$ Definition of Complex Cosecant Function $\quad$ $\displaystyle$ $=$ $\displaystyle 1 / \frac {e^{i z} - e^{-i z} } {2 i}$ $\quad$ Sine Exponential Formulation $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {2 i} {e^{i z} - e^{-i z} }$ $\quad$ multiplying top and bottom by $2 i$ $\quad$

$\blacksquare$