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

Let  $z=r_1{e}^{i\theta }\text{ and }w=r_2{e}^{i\varphi }\Rightarrow \overline{z}=r_1{e}^{-i\theta }$

Given, $\left|zw\right|=1\Rightarrow \left|r_1{e}^{i\theta }\cdot r_2{e}^{i\varphi }\right|=1$

$\Rightarrow r_1{r}_2=1$                     ... (i)

and     $\arg \left(z\right)-\arg \left(w\right)=\frac{\pi }{2}$ 

$\Rightarrow \theta -\varphi =\frac{\pi }{2}$      ... (ii)

Now,  $\overline{Z}W=r_1{e}^{-i\theta }\cdot r_2{e}^{i\varphi }=r_1{r}_2{e}^{-i⟨\theta -\varphi )}$     

$=1\cdot e^{i\pi /2}$                       [from Eqs. (i) and (ii)] 

$=\cos \frac{\pi }{2}-i\sin \frac{\pi }{2}=-i$