Werner Formulas/Cosine by Sine/Proof 2

Theorem

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

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

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

$\ds $

$=$

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

$\blacksquare$

1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$