If $\cos\;\alpha + \cos\;\beta + \cos\;\gamma = 0 = \sin \;\alpha + \sin \;\beta + \sin \;\gamma$, prove that $\cos^{2}\;\alpha + \cos^{2}\;\beta + \cos^{2}\;\gamma = \sin^{2} \alpha + \sin^{2}\beta + \sin^{2}\gamma = \large\frac{3}{2}$.

Question

This is the fifth part of the multi-part Q3.