Question

if p,q are complex roots of unity ,then prove that p^2+q^2-pq=-2

Answer 1

p,q are complex roots of unity
... P=w and q=w²

LHS
=P²+q²-pq
=(w) +(w²)²-w.w²
=w²+w⁴-w³
=w²+w.w³-w³
=w²+w.1-1. . . . (w³=1)
=w²+w-1
= - 1-1. . . (w²+w=-1)
=-2
=RHS