# Cotangent Exponential Formulation/Proof 1

$\cot z = i \dfrac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$
 $\displaystyle \cot z$ $=$ $\displaystyle \frac {\cos z} {\sin z}$ $\quad$ Definition of Complex Cotangent Function $\quad$ $\displaystyle$ $=$ $\displaystyle \frac {e^{i z} + e^{-i z} } 2 / \frac {e^{i z} - e^{-i z} } {2 i}$ $\quad$ Sine Exponential Formulation and Cosine Exponential Formulation $\quad$ $\displaystyle$ $=$ $\displaystyle i \frac {e^{i z} + e^{-i z} } {e^{i z} - e^{-i z} }$ $\quad$ multiplying numerator and denominator by $2 i$ $\quad$
$\blacksquare$