Werner Formulas/Hyperbolic Cosine by Hyperbolic Cosine

Theorem

$\cosh x \cosh y = \dfrac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$

where $\cosh$ denotes hyperbolic cosine.

$\ds $

$\ds \frac {\cosh \paren {x + y} + \cosh \paren {x - y} } 2$

$\ds $

$=$

$\ds \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \cosh \paren {x - y} } 2$

$\ds $

$=$

$\ds \frac {\paren {\cosh x \cosh y + \sinh x \sinh y} + \paren {\cosh x \cosh y - \sinh x \sinh y} } 2$

$\ds $

$=$

$\ds \frac {2 \cosh x \cosh y} 2$

$\ds $

$=$

$\ds \cosh x \cosh y$

$\blacksquare$

This result can also be seen presented as:

$2 \cosh x \cosh y = \cosh \paren {x + y} + \cosh \paren {x - y}$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $8 \ \text{(i)}$
1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.47$: Sum, Difference and Product of Hyperbolic Functions