Question

re(z1z2)=re(z1)re(z2)-im(z1)im(z2)

Answer 1

Use the trigonometric/polar expressions of \(z_1,z_2\).

Let \(\large z_1=re^{i\theta}\) and \(\large z_2=se^{i\phi}\). Then
\[\large z_1z_2=rse^{i(\theta+\phi)}=rs\bigg(\cos(\theta+\phi)+i\sin(\theta+\phi)\bigg)\]
\[\text{Re}(z_1z_2)=rs\cos(\theta+\phi)~~\text{and}~~\text{Im}(z_1z_2)=rs\sin(\theta+\phi)\]
Then, use the following identities:
\[\cos(x+y)=\cos x\cos y-\sin x\sin y\\
\sin(x+y)=\sin x\cos y+\cos x\sin y\]