Tangent of Sum of Three Angles

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Theorem

$\map \tan {A + B + C} = \dfrac {\tan A + \tan B + \tan C - \tan A \tan B \tan C} {1 - \tan B \tan C - \tan C \tan A - \tan A \tan B}$


Proof 1

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

$\blacksquare$


Proof 2

\(\ds \map \tan {A + B + C}\) \(=\) \(\ds \dfrac {\tan A + \map \tan {B + C} } {1 - \tan A \tan {B + C} }\) Tangent of Sum
\(\ds \) \(=\) \(\ds \dfrac {\tan A + \frac {\tan B + \tan C} {1 - \tan B \tan C} } {1 - \tan A \frac {\tan B + \tan C} {1 - \tan B \tan C} }\) Tangent of Sum
\(\ds \) \(=\) \(\ds \dfrac {\tan A \paren {1 - \tan B \tan C} + \tan B + \tan C} {\paren {1 - \tan B \tan C} - \tan A \paren {\tan B + \tan C} }\) multiplying top and bottom by $1 - \tan B \tan C$
\(\ds \) \(=\) \(\ds \dfrac {\tan A + \tan B + \tan C - \tan A \tan B \tan C} {1 - \tan B \tan C - \tan C \tan A - \tan A \tan B}\) simplification

$\blacksquare$


Proof 3

This is a special case of Tangent of Sum of Series of Angles, for $n = 3$.

$\blacksquare$


Also see


Sources

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