Cosine in terms of Hyperbolic Cosine

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Theorem

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

Then:

$\cos z = \map \cosh {i z}$

where:

$\cos$ denotes the complex cosine
$\cosh$ denotes the hyperbolic cosine
$i$ is the imaginary unit: $i^2 = -1$.


Proof

\(\displaystyle \map \cosh {i z}\) \(=\) \(\displaystyle \frac {e^{i z} + e^{-i z} } 2\) Definition of Hyperbolic Cosine
\(\displaystyle \) \(=\) \(\displaystyle \cos z\) Cosine Exponential Formulation

$\blacksquare$


Also see


Sources