Cotangent in terms of Hyperbolic Cotangent

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Theorem

Let $z \in \C$ be a complex number.

Then:

$i \cot z = -\coth \paren {i z}$

where:

$\cot$ denotes the cotangent function
$\coth$ denotes the hyperbolic cotangent
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\ds i \cot z\) \(=\) \(\ds i \frac {\cos z} {\sin z}\) Definition of Complex Cotangent Function
\(\ds \) \(=\) \(\ds -\frac {\cos z} {i \sin z}\) $i^2 = -1$
\(\ds \) \(=\) \(\ds -\frac {\cosh \paren {i z} } {i \sin z}\) Cosine in terms of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds -\frac {\cosh \paren {i z} } {\sinh \paren {i z} }\) Sine in terms of Hyperbolic Sine
\(\ds \) \(=\) \(\ds -\coth \paren {i z}\) Definition of Hyperbolic Cotangent

$\blacksquare$


Also see


Sources

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