Werner Formulas/Cosine by Sine/Proof 2

Theorem

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

Proof

\(\ds \cos \alpha \sin \beta\)

\(=\)

\(\ds \sin \beta \cos \alpha\)

\(\ds \)

\(=\)

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

Werner Formula for Sine by Cosine

\(\ds \)

\(=\)

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

\(\ds \)

\(=\)

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

Sine Function is Odd

$\blacksquare$

Sources

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