If $\mathrm{z}$ and $\mathrm{\omega }$ are two non-zero complex numbers such that $\left|\mathrm{z\omega }\right|=1$ and $\arg \left(\mathrm{z}\right)-\arg \left(\mathrm{\omega }\right)=\frac{\mathrm{\pi }}{2}, \mathrm{then} \overline{\mathrm{z}}\mathrm{\omega }$ is equal to

$$\begin{gathered} \left|z\omega \right|=1 \Rightarrow \left|\mathrm{z}\right| \left|\omega \right|=1 \Rightarrow \left|\overline{\mathrm{z}}\right| \left|\omega \right|=1 \therefore \left|\overline{\mathrm{z}}\omega \right|=1 \\ \mathrm{and} \mathrm{Arg} \mathrm{z}-\mathrm{Arg} \omega =\frac{\mathrm{\pi }}{2} \\ \Rightarrow -\mathrm{Arg} \mathrm{z}+\mathrm{Arg} \mathrm{w}=\frac{-\mathrm{\pi }}{2} \\ \Rightarrow \mathrm{Arg} \overline{\mathrm{z}} +\mathrm{Arg} \mathrm{w}=\frac{-\mathrm{\pi }}{2} \\ \Rightarrow \overline{\mathrm{z}} \mathrm{w}=|\overline{\mathrm{z}} \mathrm{w}| {\mathrm{e}}^{\mathrm{i} \mathrm{Arg} (\overline{\mathrm{z}} \mathrm{w})} \\ =1 {\mathrm{e}}^{-\mathrm{i} \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} =\cos \frac{\mathrm{\pi }}{2}-\mathrm{isin}\frac{\mathrm{\pi }}{2}=-\mathrm{i} \end{gathered}$$