Describe the set whose points satisfy the following relation: $|z^2 - 1| < 1$ There is a hint which states to use polar coordinates, but I feel like that complicates the problem more. As far as trying it myself, I get lost very early on. If we take $z = r(\cos{\theta} + i\sin{\theta})$, then we have
$|r^2(\cos{2\theta} + i\sin{2\theta}) - 1| < 1$
But I have no idea how to find the modulus of this point with that extra $-1$ in there.

If you use $re^{i\theta}$ as the polar representation than you get $$|r^2e^{2i\theta}-1|<1\Rightarrow
|(re^{i\theta}+1)(re^{i\theta}-1)|<1$$ which might be enlightening....
