Cotangent of Sum of Three Angles
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Theorem
- $\map \cot {A + B + C} = \dfrac {\cot A + \cot B + \cot C - \cot A \cot B \cot C} {1 - \cot B \cot C - \cot C \cot A - \cot A \cot B}$
Proof
| \(\ds \map \cos {A + B + C}\) | \(=\) | \(\ds \cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C\) | Cosine of Sum of Three Angles | |||||||||||
| \(\ds \map \sin {A + B + C}\) | \(=\) | \(\ds \sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C\) | Sine of Sum of Three Angles | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \map \cot {A + B + C}\) | \(=\) | \(\ds \dfrac {\cos A \cos B \cos C - \sin A \sin B \cos C - \sin A \cos B \sin C - \cos A \sin B \sin C} {\sin A \cos B \cos C + \cos A \sin B \cos C + \cos A \cos B \sin C - \sin A \sin B \sin C}\) | Cotangent is Cosine divided by Sine | ||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\frac {\cos A \cos B \cos C} {\sin A \sin B \sin C} - \frac {\sin A \sin B \cos C} {\sin A \sin B \sin C} - \frac {\sin A \cos B \sin C} {\sin A \sin B \sin C} - \frac {\cos A \sin B \sin C} {\sin A \sin B \sin C} } {\frac {\sin A \cos B \cos C} {\sin A \sin B \sin C} + \frac {\cos A \sin B \cos C} {\sin A \sin B \sin C} + \frac {\cos A \cos B \sin C} {\sin A \sin B \sin C} - \frac {\sin A \sin B \sin C} {\sin A \sin B \sin C} }\) | dividing numerator and denominator by $\sin A \sin B \sin C$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {\cot A + \cot B + \cot C - \cot A \cot B \cot C} {1 - \cot B \cot C - \cot C \cot A - \cot A \cot B}\) | Cotangent is Cosine divided by Sine and simplifying |
$\blacksquare$