Werner Formulas/Cosine by Sine/Proof 1

Theorem

$\cos \alpha \sin \beta = \dfrac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$

$\ds $

$\ds \frac {\map \sin {\alpha + \beta} - \map \sin {\alpha - \beta} } 2$

$\ds $

$=$

$\ds \frac {\paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta} - \paren {\sin \alpha \cos \beta - \cos \alpha \sin \beta} } 2$

$\ds $

$=$

$\ds \frac {2 \cos \alpha \sin \beta} 2$

$\ds $

$=$

$\ds \cos \alpha \sin \beta$

$\blacksquare$