Tangent of Sum/Proof
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Theorem
- $\map \tan {a + b} = \dfrac {\tan a + \tan b} {1 - \tan a \tan b}$
Proof
\(\ds \map \tan {a + b}\) | \(=\) | \(\ds \frac {\map \sin {a + b} } {\map \cos {a + b} }\) | Tangent is Sine divided by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sin a \cos b + \cos a \sin b} {\cos a \cos b - \sin a \sin b}\) | Sine of Sum and Cosine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\frac {\sin a} {\cos a} + \frac {\sin b} {\cos b} } {1 - \frac {\sin a \sin b} {\cos a \cos b} }\) | dividing top and bottom by $\cos a \cos b$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\tan a + \tan b} {1 - \tan a \tan b}\) | Tangent is Sine divided by Cosine |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Tangents of sum and difference