If [tex] z_1 = a+bi [/tex] and [tex]z_2=c+di[/tex], then
[tex]z_1 z_2 = (a+bi)(c+di) = ac+adi+bci+bdi^2 = (ac-bd)+(ad+bc)i[/tex]
So [tex]\overline{z_1 z_2} = (ac-bd) - (ad+bc)i[/tex].
Next, compute [tex]\overline{z_1} \cdot \overline{z_2}[/tex].
[tex]\overline{z_1} \cdot \overline{z_2} = (a-bi)(c-di) [/tex]
[tex]= ac-adi-bci+bdi^2 = (ac-bd) -(ad+bc)i [/tex]
