F `a=cos(2pi//7)+is in(2pi//7)` , then find the quadratic equation whose roots are `alpha=a=a^2+a^4a n dbeta=a^3=a^5+a^7` .

Explanation

`a=cos(2pi//7)+isin(2pi//7)`, which is one of the 7th roots of unity.
Therefore, seventh roots of units are `1,a,a^(2),a^(3),a^(4),a^(5)" and "a^(6).`
Also, `a^(7)=1" "...(1)`
Sum of the roots `=1+a+a^(2)+a^(3)+a^(4)+a^(5)+a^(6)=0`
`:." "S=alpha+beta=(a+a^(2)+a^(4))+(a^(3)+a^(5)+a^(6))=-1`
Product of roots,
`P=alphabeta=(a+a^(2)+a^(4))(a^(3)+a^(5)+a^(6))`
`=a^(4)+a^(6)+a^(7)+a^(5)+a^(7)+a^(8)+a^(7)+a^(9)+a^(10)`
`=a^(4)+a^(6)+a+a^(5)+1+a+1+a^(2)+a^(3)" "["from Eq. (1)"]`
`=3+(a+a^(2)+a^(3)+a^(4)+a^(5)+a^(6))`
`=3-1=2`
Therefore, required equation is : `x^(2)-Sx+P=0" or "x^(2)+x+2=0`