Complex number z: |z| = 1 => z* = 1/z proof

Question

I can t understand how we move from $ |z| =1 \rightarrow z* = \frac{1}{z}, \quad z \in C $ ($z*$ is the complex conjugate)

Answer 1

Hint: square both sides of $|z|=1$ and use $|z|^2={z\bar z}$

Answer 2

$$|z|=1\iff|z|^2=1\iff z\bar{z}=1\iff\bar{z}=\frac1z$$