Question

Could someone show me step by step how to prove the product and quotient of two complex numbers in trig form

Answer 1

you know \[z = (x + i y) = R e^{i \theta} = R(\cos \theta + i \sin \theta), R = |z|\] right?

this allows you to do all sorts of things in shorthand, including multiplying or dividing numbers by adding or subtracting the thetas of the complex numbers

apply this to the specific proof you have in mind.