Find all complex numbers $z$ such that $| z | = 1$ and $ |\frac z {\overline z} + \frac {\bar z} z | = 1. $ [closed]

Generalization:

Let $z=r(\cos t+i\sin t)$ where $r>0$ and $t$ is real

$\dfrac z{\bar z}=\cos2t+i\sin2t$

$\iff \dfrac{\bar z}z=\cos2t-i\sin2t$

$\implies|2\cos2t|=1\implies\cos2t=\pm\dfrac12$

$\implies\cos4t=\cdots=-\dfrac12=\cos120^\circ$

$\implies4t=360^\circ n\pm120^\circ$ where $n$ is any integer

$\iff t=?$ where $0\le n\le3$

So, we have $4+4$ in-congruent solutions

Much like @Lab Bhattacharjee’s answer, but cast in geometric language.

Zero’th, if $w$ is a complex number, then $w+\overline w$ is real, twice the real part of $w$.

First, to require $|z|=1$ is to put $z$ on the unit circle.

Second, for points $z$ on the unit circle, $\overline z=1/z$, so that for our points, $z/\overline z=z^2$ and $\overline z/z={\overline z}^2=\overline{z^2}$.

Third, the second condition therefore says that $2\Re(z^2)=1$ or $2\Re(z^2)=-1$.

If we call $\xi$ the primitive twelfth root of unity $\cos\frac\pi6+i\sin\frac\pi6$, then the first possibility in (3) above is $z=\pm\xi$ or $z=\pm\xi^{11}$; the second possibility above is $\xi=\pm\xi^2$ or $\pm\xi^4$.

In other words, $z$, as a number on the clock-face of the unit circle, lies at hours 1,2,4,5,7,8,10,11.