Simpson's Formulas/Sine by Cosine

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 $\displaystyle$ $=$ $\displaystyle \frac 1 {2 i} \paren {\exp \paren {i \alpha} - \exp \paren {-i \alpha} } \paren {\exp \paren {i \beta} + \exp \paren {-i \beta} }$ $\displaystyle$ $=$ $\displaystyle \frac 1 {2 i} \paren {\exp \paren {i \paren {\alpha + \beta} } - \exp \paren {-i \paren {\alpha + \beta} } + \exp \paren {i \paren {\alpha - \beta} } - \exp \paren {-i \paren {\alpha - \beta} } }$ $\displaystyle$ $=$ $\displaystyle \frac {\exp \paren {i \paren {\alpha + \beta} } - \exp \paren {-i \paren {\alpha + \beta} } } {2 i} + \frac {\exp \paren {i \paren {\alpha - \beta} } - \exp \paren {-i \paren {\alpha - \beta} } } {2 i}$ $\displaystyle$ $=$ $\displaystyle \sin \paren {\alpha + \beta} + \sin \paren {\alpha - \beta}$

$\blacksquare$