Cosine minus Cosine/Proof 2
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Theorem
- $\cos \alpha - \cos \beta = -2 \, \map \sin {\dfrac {\alpha + \beta} 2} \, \map \sin {\dfrac {\alpha - \beta} 2}$
Proof
| \(\ds \) | \(\) | \(\ds -2 \map \sin {\frac {\alpha + \beta} 2} \map \sin {\frac {\alpha - \beta} 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds -2 \frac {\map \cos {\dfrac {\alpha + \beta} 2 - \dfrac {\alpha - \beta} 2} - \map \cos {\dfrac {\alpha + \beta} 2 + \dfrac {\alpha - \beta} 2} } 2\) | Werner Formula for Sine by Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\paren {\cos \frac {2 \beta} 2 - \cos \frac {2 \alpha} 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \alpha - \cos \beta\) |
$\blacksquare$