Cosinus einer Summe oder Differenz

Summe

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

   →   Beweis

   $cos(\alpha +\beta)$    $=\ \frac{OA}{OC}$    $\ =\ \frac{OB}{OC}\ -\ \frac{AB}{OC}$    $\ ==\ \frac{OB}{OC}\ -\ \frac{DE}{OC}$    $\ =\ \frac{OB}{OE}\frac{OE}{OC}-\frac{DE}{CE}\frac{CE}{OC}$    $\ =cos(\alpha) cos(\beta)\ -\ sin(\alpha)sin(\beta)$

<Differenz

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

   →   Beweis

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