Past exam of the mathematical methods I course of the University of Oxford / 2022 / 9 / a / iii / Solution

For $z\ne -1$, multiply numerator and denominator by $\overline{z}+1$. The real part isThus its zero locus isIt is the unit circle with the point $-1$ removed. That point cannot be restored by the zero numerator obtained after clearing a denominator: the original expression is undefined there.