Tangent in terms of Hyperbolic Tangent

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Theorem

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

Then:

$i \tan z = \map \tanh {i z}$

where:

$\tan$ denotes the tangent function
$\tanh$ denotes the hyperbolic tangent
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\ds \map \tanh {i z}\) \(=\) \(\ds \frac {\map \sinh {i z} } {\map \cosh {i z} }\) Definition of Hyperbolic Tangent
\(\ds \) \(=\) \(\ds \frac {i \sin z} {\map \cosh {i z} }\) Sine in terms of Hyperbolic Sine
\(\ds \) \(=\) \(\ds \frac {i \sin z} {\cos z}\) Cosine in terms of Hyperbolic Cosine
\(\ds \) \(=\) \(\ds i \tan z\) Definition of Complex Tangent Function

$\blacksquare$


Also see


Sources

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