Well, actually, there are 4 primitive 8th roots of unity ( ), so presumably your proof should work for any one of them.
Let be a primitive 8th root of unity. Note that
so .
(If we had any other of the factors = 0, say , then would be an 8th root of unity but not a primitive root. Or, to put it another way, is the 8th cyclotomic polynomial.)
Since is a field, .
And
so , hence .