If $ \cos\;\alpha + \cos\;\beta + \cos\;\gamma = 0 = \sin \;\alpha + \sin \;\beta + \sin \;\gamma $, prove that $\sin 2\alpha + \sin 2\beta + \sin 2\gamma = 0$

Question

(Hints: Take $a = \textit{$\cos$} \;\alpha, \;b = \textit{$\cos$} \;\beta, c = \textit{$\cos$} \; \gamma$, $a+b+c = 0 => a^{3} + b^{3}+ c^{3} = 3abc$ $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = 0 => a^{2}+b^{2}+c^{2}=0$ This is the fourth part of the multi-part Q3.