Werner Formulas/Sine by Cosine/Proof 2

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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

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