Cotangent of Difference

Corollary to Cotangent of Sum

$\map \cot {a - b} = \dfrac {\cot a \cot b + 1} {\cot b - \cot a}$

where $\cot $ is cotangent.


Proof

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

$\blacksquare$


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.37$
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