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