Find all complex numbers $z$ satisfying $|z| = 1$ and $\left| \tfrac{z}{\bar{z}} + \tfrac{\bar{z}}{z} \right| = 1$ 
Find all complex numbers $z$ satisfying the following conditions: $$|z| = 1 \qquad \text{and} \qquad \left| \dfrac{z}{\bar{z}} + \dfrac{\bar{z}}{z} \right| = 1 .$$ Here, $|\cdot|$ means the norm.

On the right side I do,
$$ |z^2 + \bar{z}^2|= z\bar{z}$$
but I don't know how to continue.

Hint
Since $z\bar z=|z|^2$, your equation is equivalent at
$$|z^2+\bar z^2|=1$$
