Prosthaphaeresis Formulas/Hyperbolic Sine minus Hyperbolic Sine
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Theorem
- $\sinh x - \sinh y = 2 \map \cosh {\dfrac {x + y} 2} \map \sinh {\dfrac {x - y} 2}$
where $\sinh$ denotes hyperbolic sine and $\cosh$ denotes hyperbolic cosine.
Proof
\(\ds \) | \(\) | \(\ds 2 \map \cosh {\frac {x + y} 2} \map \sinh {\frac {x - y} 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \frac {\map \sinh {\dfrac {x - y} 2 + \dfrac {x + y} 2} + \map \sinh {\dfrac {x - y} 2 - \dfrac {x + y} 2} } 2\) | Werner Formula for Hyperbolic Sine by Hyperbolic Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \sinh \frac {2 x} 2 + \map \sinh {-\frac {2 y} 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sinh \frac {2 x} 2 - \sinh \frac {2 y} 2\) | Hyperbolic Sine Function is Odd | |||||||||||
\(\ds \) | \(=\) | \(\ds \sinh x - \sinh y\) |
$\blacksquare$
Linguistic Note
The word prosthaphaeresis or prosthapheiresis is a neologism coined some time in the $16$th century from the two Greek words:
- prosthesis, meaning addition
- aphaeresis or apheiresis, meaning subtraction.
With the advent of machines to aid the process of arithmetic, this word now has only historical significance.
Ian Stewart, in his Taming the Infinite from $2008$, accurately and somewhat diplomatically describes the word as "ungainly".
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.43$: Sum, Difference and Product of Hyperbolic Functions