Hyperbolic Cotangent of Sum
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Theorem
- $\map \coth {a + b} = \dfrac {\coth a \coth b + 1} {\coth b + \coth a}$
where $\coth$ denotes the hyperbolic cotangent.
Corollary
- $\map \coth {a - b} = \dfrac {\coth a \coth b - 1} {\coth b - \coth a}$
Proof
\(\ds \map \coth {a + b}\) | \(=\) | \(\ds \frac {\map \cosh {a + b} } {\map \sinh {a + b} }\) | Definition 2 of Hyperbolic Cotangent | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\cosh a \cosh b + \sinh a \sinh b} {\sinh a \cosh b + \cosh a \sinh b}\) | Hyperbolic Sine of Sum and Hyperbolic Cosine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac {\cosh a \cosh b} {\sinh a \sinh b} + 1} {\frac {\cosh b} {\sinh b} + \frac {\cosh a} {\sinh a} }\) | dividing the numerator and denominator by $\sinh a \sinh b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\coth a \coth b + 1} {\coth b + \coth a}\) | Definition 2 of Hyperbolic Cotangent |
$\blacksquare$
Also see
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 8$: Hyperbolic Functions: $8.23$: Addition Formulas