Cotangent of i/Proof 1

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Theorem

$\cot i = \paren {\dfrac {1 + e^2} {1 - e^2} } i$


Proof

\(\ds \cot i\) \(=\) \(\ds \frac {\cos i} {\sin i}\) Definition of Complex Cotangent Function
\(\ds \) \(=\) \(\ds \frac {\frac e 2 + \frac 1 {2 e} } {\left({\frac e 2 - \frac 1 {2 e} }\right) i}\) Cosine of $i$ and Sine of $i$
\(\ds \) \(=\) \(\ds \left({\frac {e + \frac 1 e} {e - \frac 1 e} }\right) \left({\frac 1 i}\right)\) multiplying denominator and numerator by $2$
\(\ds \) \(=\) \(\ds \left({ \frac {e^2 + 1} {e^2 - 1} }\right) \left({\frac 1 i}\right)\) multiplying denominator and numerator by $e$
\(\ds \) \(=\) \(\ds \left({ \frac {1 + e^2} {1 - e^2} }\right) i\) Reciprocal of $i$

$\blacksquare$