(related to Proposition: Multiplication of Complex Numbers Using Polar Coordinates)

- Because the polar coordinates uniquely represent the complex numbers, it suffices to prove the given formula.
- Let $z=r\exp(i\phi)$ and $w=s\exp(i\psi)$ be two complex numbers.
- For their product it follows by the additivity theorems for cosine and sine:

$$\begin{array}{rcl}z\cdot w&=&r\exp(i\phi)\cdot s\exp(i\psi)\\ &=&r(\cos(\phi)+i\sin(\phi))\cdot s(\cos(\psi)+i\sin(\psi))\\ &=& (r\cdot s)\cdot [\cos(\phi)\cdot\cos(\psi)+i\sin(\phi)\cdot\sin(\psi) + i\sin(\psi)\cos(\phi)-\sin(\phi)\sin(\psi)]\\ &=& (r\cdot s)\cdot [\cos(\phi+\psi)+i\sin(\phi+\psi)]. \end{array}$$

∎

**Modler, F.; Kreh, M.**: "Tutorium Analysis 1 und Lineare Algebra 1", Springer Spektrum, 2018, 4th Edition