# 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 $n = 3$.

$\blacksquare$