Proof of trigonometric multiplication of complex numbers

[embassyhill]

This is supposed to be a proof of trigonometric multiplication of complex numbers:

What happened at the =...= point? I understand everything up to that.

 

[Upisoft]

This is supposed to be a proof of trigonometric multiplication of complex numbers:

What happened at the =...= point? I understand everything up to that.

Do the multiplication and see what happens. Bottom part is just equal to r.

 

[embassyhill]

I did manage to get cos(alpha-beta)+isin(alpha-beta) on the upper part but I was too dumb to remember cos^2+sin^2=1 :P. Thanks.