Complex equation: $z^8 = (1+z^2)^4$ What's up with this complex equation?
$ z^8 = (1+z^2)^4 $
To start with, there seems to be a problem when we try to apply root of four to both sides of the equation:
$ z^8 = (1+z^2)^4 $
$ z^2 = 1 + z^2 $
which very clearly doesn't have any solutions, but we know there are solutions: the problem is from an exam, and, besides, wolphram alpha gladily gives them to us.
We've tried to solve it using the trigonomectric form, but the sum inside of the parenthesis is killing all of our attempts.
Any help? Ideas?

$$\displaystyle z^8 = (1+z^2)^4 \Rightarrow 1=(1+\frac{1}{z^2})^4$$
Now find 4-th root of unity.
Thus $\displaystyle (1+\frac{1}{z^2})=1,i,-1,-i$
Now solve for $z^2$.
