Answer
See below.
Work Step by Step
Let $z_1=a+bi$ and $z_2=c+di$. Then $\overline{z_1z_2}=\overline{(a+bi)(c+di)}=\overline{ac+adi+bci-bd}=(ac-bd)-(bc+ad)i$
$\overline{z_1}\cdot\overline{z_2}=\overline{(a+bi)}\cdot\overline{(c+di)}=(a-bi)(c-di)=ac-adi-bci-bd$. Thus they are equal, thus we proved what we wanted to.