Cotangent of Difference
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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: $5.37$