The value of $\sin \left[{\log }_{\mathrm{e}} \left\{{\left(\cos \frac{\mathrm{\pi }}{2}+\mathrm{isin} \frac{\mathrm{\pi }}{2}\right)}^{\mathrm{z}}\right\}\right]$ is, where z satisfies the equation $|\mathrm{z}-2\mathrm{i}|=1$ and has least modulus

$|\mathrm{z}-2\mathrm{i}|=1$ is a circle with centre C (0, 2) and r  = 1 

z= complex number  of least modulus which lies on $|\mathrm{z}-2\mathrm{i}|=1$ $\mathrm{is} P(0,1)=\mathrm{i}={\mathrm{e}}^{\mathrm{i\pi }/2}$
$$\begin{gathered} {\left(\cos \frac{\mathrm{\pi }}{2}+\mathrm{isin}\frac{\mathrm{\pi }}{2}\right)}^z={\mathrm{i}}^{\mathrm{i}}={\left({\mathrm{e}}^{\mathrm{i\pi }/2}\right)}^i={\mathrm{e}}^{-\mathrm{\pi }/2} \\ \sin \left[{\log }_{\mathrm{e}} \left\{{\left(\cos \frac{\mathrm{\pi }}{2}+\mathrm{isin} \frac{\mathrm{\pi }}{2}\right)}^{\mathrm{z}}\right\}\right]\Rightarrow \sin \left[{\log }_e{e}^{-\mathrm{\pi }/2}\right]=\sin \left(\frac{-\mathrm{\pi }}{2}\right)=-1 \end{gathered}$$