The number of complex numbers z, satisfying |z| = 1and $\left|\frac{\mathrm{z}}{\overline{\mathrm{z}}}+\frac{\overline{\mathrm{z}}}{\mathrm{z}}\right|=1,$is : 

$\left|\frac{z}{\overline{z}}+\frac{\overline{z}}{z}\right|=1$ and $\left|z\right|=1$

$\left|z^2+{\left(\overline{z}\right)}^2\right|=1$

Let $z=x+iy$

$\left|{\left(x+iy\right)}^2+{\left(x-iy\right)}^2\right|=1$

$\left|x^2-y^2\right|=\frac{1}{2}$

$x^2+y^2=1 and x^2-y^2=\pm \frac{1}{2}$

$\left(x,y\right)=\left(\pm \frac{\sqrt{3}}{2},\pm \frac{1}{2}\right) and \left(\pm \frac{1}{2}, \pm \frac{\sqrt{3}}{2}\right)$
8 solution