Werner Formulas/Sine by Cosine/Proof 2

Theorem

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

Proof

$\ds $

$\ds 2 \sin \alpha \cos \beta$

$\ds $

$=$

$\ds 2 \paren {\dfrac {\map \exp {i \alpha} - \map \exp {-i \alpha} } {2 i} } \paren {\dfrac {\map \exp {i \beta} + \map \exp {-i \beta} } 2}$

Euler's Sine Identity and Euler's Cosine Identity

$\ds $

$=$

$\ds \frac 1 {2 i} \paren {\map \exp {i \alpha} - \map \exp {-i \alpha} } \paren {\map \exp {i \beta} + \map \exp {-i \beta} }$

$\ds $

$=$

$\ds \frac 1 {2 i} \paren {\map \exp {i \paren {\alpha + \beta} } - \map \exp {-i \paren {\alpha + \beta} } + \map \exp {i \paren {\alpha - \beta} } - \map \exp {-i \paren {\alpha - \beta} } }$

$\ds $

$=$

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

$\ds $

$=$

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

$\blacksquare$

Sources

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