absolute value - Complex number with z to the power of 4

I have to find all $z\in C$ for which BOTH of the following is true:

1) $|z|=1$

2) $|z^4+1| = 1$

I understand that the 1) is a unit circle, but I can't find out what would be the 2).

Calculating the power of $(x+yi)$ seems wrong because I get a function with square root of polynomial of the 8th degree.

I've even tried to equate the two, since their modulo is the same, like this:

$|z|= |z^4+1|$

and then separating two possibilities in which the equation holds, but I get only

$z^4\pm z+1=0$

which I don't know how to calculate.

I guess there is an elegant solution which I am unable to see...

Any help?

Answer

It seems helpful to square both sides. From the second equation, you get

$$1 = (z^4 + 1)(\overline{z}^4 + 1) = (z \overline{z})^4 + z^4 + \overline{z}^4 + 1 = z^4 + \overline{z}^4 + 2.$$ That is, $z^4 + \overline{z^4} = -1.$

The first equation lets you write $z = e^{i \theta}$, so you have

$$-1 = e^{4i\theta} + e^{-4i \theta} = 2\cos(4\theta);$$ in other words, $\cos(4\theta) = \frac{-1}{2}.$ The solutions of this are well-known.