`(1)/(a + omega) + (1)/(b+omega) +(1)/(c + omega) + (1)/(d + omega) =(1)/(omega)` where, a,b,c,d, `in` R and `omega` is a complex cube root of unity then find the value of `sum (1)/(a^(2)-a+1)`

Explanation

`(1)/(a+ omega) + (1)/(b+ omega) + (1)/(c+omega)+(1)/(d+omega) = (1)/(omega)" "(1)`
Taking conjugate on both sides, we get
`(1)/(a+omega^(2)) + (1)/(b+omega^(2)) +(1)/(c+omega^(2)) + (1)/(d+omega^(2)) = (1)/(omega^(2))" "(2)`
Subtracting, we get
`sum((1)/(a+omega^(2))-(1)/(a+omega^(2)))=(1)/(omega) -(1)/(omega^(2))`
`rArr (omega^(2)-omega) sum (1)/((a+omega)(a+omega^(2))) = omega^(2) - omega`
`rArr sum(1)/(a^(2) -a+1)=1`