# Tangent Exponential Formulation

## Theorem

Let $z$ be a complex number.

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

Then:

### Formulation 1

$\tan z = i \dfrac {1 - e^{2 i z} } {1 + e^{2 i z} }$

### Formulation 2

$\tan z = \dfrac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }$

### Formulation 3

$\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$