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cos A + cos B + cos C=0 ; i*(sin A + sin B + sin C)=0
cos A + i sin A + cos B + i sin B + cos C + i sin C=0
Z1 + Z2 + Z3 = 0
now use this identity
a^3 + b^3 + c^3 - 3abc = (a+b+c) (a^2 + b^2 + c^2 - ab - bc - ac)
Z1^3 + Z2^3 + Z3^3 - a Z1 Z2 Z3 = 0
cos 3A + i sin 3A + cos 3B + i sin 3B + cos 3C + i sin 3C = 3 }cos (A+B+C) + i sin (A+B+C)}
now compare real and imaginary parts.
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