Werner Formulas/Sine by Cosine/Proof 1

Theorem

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

$\ds $

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

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

$\blacksquare$