Sum of Hyperbolic Sine and Cosine equals Exponential

Theorem

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

Then:

$e^z = \cosh z + \sinh z$

where:

$e^z$ denotes the complex exponential function

$\cosh z$ denotes the cosine function

$\sinh z$ denotes sine function

Proof

\(\ds \cosh z + \sinh z\)

\(=\)

\(\ds \dfrac {e^z + e^{-z} } 2 + \dfrac {e^z - e^{-z} } 2\)

Definition of Hyperbolic Cosine and Definition of Hyperbolic Cosine

\(\ds \)

\(=\)

\(\ds \dfrac {e^z + e^z + e^{-z} - e^{-z} } 2\)

\(\ds \)

\(=\)

\(\ds e^z\)

$\blacksquare$

Also see

• Euler's Formula

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions