Question

If $a=e^{i\alpha },b=e^{i\beta },c=e^{i\gamma }$ and $\cos \alpha +\cos \beta +\cos \gamma =0=\sin \alpha +\sin \beta +\sin \gamma$, then prove the following
$\sum \cos 2\alpha =0=\sum \sin 2\alpha$

Answer 1

$a=e^{i\alpha },b=e^{i\beta },c=e^{i\gamma }$

$\cos \alpha +\cos \beta +\cos \gamma =\sin \alpha +\sin \beta +\sin \gamma =0$

$⟹\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$

$a+b+c=0⟹a^2+b^2+c^2+2(ab+bc+ca)=0⟹a^2+b^2+c^2=0$

$\therefore \sum \cos 2\alpha =\sum \sin 2\alpha =0$