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: $5.37$