2009/010) Prove Cos pi/7 + cos 3pi/7 + cos 5pi/7=1/2

To prove
Cos pi/7 + cos 3pi/7 + cos 5pi/7=1/2

let z = cos pi/7 + i sin pi/7

z^7 = cos pi + i sin pi = - 1

so z^7+1 = 0
(z+1) (z^6-z^5+z^4 - z^3+z^2 -z + 1) = 0

as z is not - 1 so

z^6-z^5+z^4 - z^3+z^2 -z + 1 = 0

so z^6-z^5+z^4 - z^3+z^2 -z= -1

so z+z^3+ z^5 = 1 + (z^2+z^4+z6)
equating the real part

cos pi/7 + cos 3pi/7 + cos 5pi/ 7 = -1 - (cos 2pi/7+ cos 4pi/7 + cos 6pi/7)

but cos 6pi/7 = - cos(pi-6pi/7) = - cos pi/7
cos 4pi/7 = - cos 3pi/7
cos 2pi/7 = cos 5pi7

so
cos pi/7 + cos 3pi/7 + cos 5pi/ 7= 1 - (\cos pi/7 + cos 3pi/7 + cos 5pi/ 7)

or 2 (cos pi/7 + cos 3pi/7 + cos 5pi/ 7) = 1
or cos pi/7 + cos 3pi/7 + cos 5pi/ 7 = 1/2