Simpson's Formulas/Sine by Cosine

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Theorem

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

where $\sin$ denotes sine and $\cos$ denotes cosine.


Proof 1

\(\displaystyle \) \(\) \(\displaystyle \frac {\sin \paren {\alpha + \beta} + \sin \paren {\alpha - \beta} } 2\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\paren {\sin \alpha \cos \beta + \cos \alpha \sin \beta} + \paren {\sin \alpha \cos \beta - \cos \alpha \sin \beta} } 2\) Sine of Sum and Sine of Difference
\(\displaystyle \) \(=\) \(\displaystyle \frac {2 \sin \alpha \cos \beta} 2\)
\(\displaystyle \) \(=\) \(\displaystyle \sin \alpha \cos \beta\)

$\blacksquare$


Proof 2

\(\displaystyle \) \(\) \(\displaystyle 2 \sin \alpha \cos \beta\)
\(\displaystyle \) \(=\) \(\displaystyle 2 \paren {\dfrac {\exp \paren {i \alpha} - \exp \paren {-i \alpha} } {2 i} } \paren {\dfrac {\exp \paren {i \beta} + \exp \paren {-i \beta} } 2}\) Sine Exponential Formulation and Cosine Exponential Formulation
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \paren {\exp \paren {i \alpha} - \exp \paren {-i \alpha} } \paren {\exp \paren {i \beta} + \exp \paren {-i \beta} }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac 1 {2 i} \paren {\exp \paren {i \paren {\alpha + \beta} } - \exp \paren {-i \paren {\alpha + \beta} } + \exp \paren {i \paren {\alpha - \beta} } - \exp \paren {-i \paren {\alpha - \beta} } }\)
\(\displaystyle \) \(=\) \(\displaystyle \frac {\exp \paren {i \paren {\alpha + \beta} } - \exp \paren {-i \paren {\alpha + \beta} } } {2 i} + \frac {\exp \paren {i \paren {\alpha - \beta} } - \exp \paren {-i \paren {\alpha - \beta} } } {2 i}\)
\(\displaystyle \) \(=\) \(\displaystyle \sin \paren {\alpha + \beta} + \sin \paren {\alpha - \beta}\)

$\blacksquare$


Sources