Hyperbolic Cotangent of Sum/Corollary

Corollary to Hyperbolic Cotangent of Sum

$\map \coth {a - b} = \dfrac {\coth a \coth b - 1} {\coth b - \coth a}$

where $\coth$ denotes the hyperbolic cotangent.


Proof

\(\ds \map \coth {a - b}\) \(=\) \(\ds \frac {\coth a \map \coth {-b} + 1} {\map \coth {-b} + \coth a}\) Hyperbolic Cotangent of Sum
\(\ds \) \(=\) \(\ds \frac {-\coth a \coth b + 1} {\coth a - \coth b}\) Hyperbolic Cotangent Function is Odd
\(\ds \) \(=\) \(\ds \frac {\coth a \coth b - 1} {\coth b - \coth a}\) multiplying numerator and denominator by $-1$ and rearranging

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.23$: Addition Formulas
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