if cos(y-z) + cos(z-x)+ cos(x-y) = -3/2, prove that cosx + cosy +cosz = 0 = sinx + siny+ sinz

cos (x-y) + cos (y-z) + cos (z-x) = -3/2
=3+2(cos (x-y) + cos (y-z) + cos (z-x))=0
=1+1+1+2(cos (x-y) + cos (y-z) + cos (z-x))=0
=(sin²x + cos²x) + (sin²y + cos²y) + (sin²z + cos²z) + 2(cosx cosy + sinx siny + cosy cosz + siny sinz + cosz cosx + sinz sinx)=0
=(sin²x + sin²y + sin²z+2(sinx siny + siny sinz + sinz sinx))+(cos²x + cos²y + cos²z +2(cosx cosy + cosy cosz + cosz cosx))=0
=(sinx + siny + sinz)² + (cosx + cosy + cosz)²=0
only and only iff
sinx + siny + sinz=0
cosx + cosy + cosz=0