If z and  w are two complex numbers such that $\left|zw\right|=1$ and $\arg \left(z\right)-\arg \left(w\right)=\pi /2$, then :

Given, $\left|zw\right|=1\Rightarrow \left|z\right|\left|w\right|=1$

Let $\left|z\right|=r\Rightarrow z=r{e}^{i\theta }$

$\therefore \left|w\right|=\frac{1}{r}\Rightarrow w=\frac{1}{r}{e}^{i\varphi }$

Now, $\arg \left(z\right)-\arg \left(w\right)=\frac{\pi }{2}$                                            [Given]

$\Rightarrow \theta -\varphi =\frac{\pi }{2}\Rightarrow \theta =\frac{\pi }{2}+\varphi$

Consider $z\overline{w}=r{e}^{i\theta }\frac{1}{r}{e}^{-i\varphi }=e^{i\left(\frac{\pi }{2}+\varphi \right)}{e}^{-i\varphi }$

$=i^{\left(\frac{\pi }{2}+\varphi -\varphi \right)}=i^{12}=i$

And $zw=r{e}^{-i\theta }\frac{1}{r}{e}^{i\varphi }=c^{-i\theta }\cdot e^{i\varphi }=e^{-i(\theta -\varphi )}=e^{-i\frac{\pi }{2}}=-i$