Cotangent Exponential Formulation/Proof 1

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Theorem

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


Proof

\(\displaystyle \cot z\) \(=\) \(\displaystyle \frac {\cos z} {\sin z}\) $\quad$ 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 $2 i$ $\quad$

$\blacksquare$