Euler's Secant Identity

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Theorem

$\sec z = \dfrac 2 {e^{i z} + e^{-i z} }$

where:

$z \in \C$ is a complex number
$\sec z$ denotes the secant function
$i$ denotes the imaginary unit: $i^2 = -1$


Proof

\(\ds \sec z\) \(=\) \(\ds \frac 1 {\cos z}\) Definition of Complex Secant Function
\(\ds \) \(=\) \(\ds 1 / \frac {e^{i z} + e^{-i z} } 2\) Euler's Sine Identity and Euler's Cosine Identity
\(\ds \) \(=\) \(\ds \frac 2 {e^{i z} + e^{-i z} }\) multiplying top and bottom by $2$

$\blacksquare$


Also see


Source of Name

This entry was named for Leonhard Paul Euler.


Sources

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