Hyperbolic Cotangent of Sum/Corollary
< Hyperbolic Cotangent of Sum(Redirected from Hyperbolic Cotangent of Difference)
Jump to navigation
Jump to search
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