Cosine minus Cosine/Proof 2

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
\(\ds \) \(=\) \(\ds -\paren {\cos \frac {2 \beta} 2 - \cos \frac {2 \alpha} 2}\)
\(\ds \) \(=\) \(\ds \cos \alpha - \cos \beta\)

$\blacksquare$

This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.