Werner Formulas/Hyperbolic Sine by Hyperbolic Cosine
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Theorem
- $\sinh x \cosh y = \dfrac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2$
where $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.
Proof
\(\ds \) | \(\) | \(\ds \frac {\sinh \paren {x + y} + \sinh \paren {x - y} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \cosh \paren {x - y} } 2\) | Hyperbolic Sine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\paren {\sinh x \cosh y + \cosh x \sinh y} + \paren {\sinh x \cosh y - \cosh x \sinh y} } 2\) | Hyperbolic Sine of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {2 \sinh x \cosh y} 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sinh x \cosh y\) |
$\blacksquare$
Also presented as
This result can also be seen presented as:
- $2 \sinh x \cosh y = \sinh \paren {x + y} + \sinh \paren {x - y}$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4$. Elementary Functions of a Complex Variable: Exercise $8 \ \text{(ii)}$
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.48$: Sum, Difference and Product of Hyperbolic Functions