Sine of a sum or difference

Sum

   $sin(\alpha\ +\ \beta)\ =\ sin(\alpha) cos(\beta)\ +\ cos(\alpha)sin(\beta)$

   →   Demonstration

    Formulas in $\frac{\pi}{2}$:→   see    $sin(\alpha +\beta)$    $\ =\ cos(\frac{\pi}{2}\ -\ (\alpha\ +\ \beta))$    $\ =\ cos((\frac{\pi}{2}\ -\ \alpha)\ -\ \beta))$    Cosine of a difference: →   see    $\ =\ cos(\frac{\pi}{2}\ -\ \alpha)cos(\beta)\ +\ sin(\beta)sin(\frac{\pi}{2}\ -\ \alpha)) $        $\ =\ sin(\alpha)cos(\beta)\ +\ sin(\beta)cos(\alpha) $

<Difference

   $sin(\alpha\ -\ \beta)\ =\ sin(\alpha) cos(\beta)\ -\ cos(\alpha)sin(\beta)$

   →   Demonstration

    $sin(\alpha\ +\ (-\beta))$     $=\ sin(\alpha) cos(-\beta)\ +\ cos(\alpha)sin(-\beta)$ ( sine of sum!)     $=\ sin(\alpha) cos(\beta)\ -\ cos(\alpha)sin(\beta)$