Multiplying complex numbers in polar form?

Could someone explain why you multiply the lengths and add the angles when multiplying polar coordinates?

I tried multiplying the polar forms ($r_1\left(\cos\theta_1 + i\sin\theta_1\right)\cdot r_2\left(\cos\theta_2 + i\sin\theta_2\right)$), and expanding/factoring the result, and end up multiplying the lengths but can't seem to come to an equation where you add the angles.

Answer

By multiplying things out as usual, you get

$$[r_1(\cos\theta_1 + i\sin\theta_1)][r_2(\cos\theta_2 + i\sin\theta_2)] = r_1r_2(\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2 + i[\sin\theta_1\cos\theta_2 + \sin\theta_2\cos\theta_1]).$$

Now you want to use the trig identities $\cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2 = \cos(\theta_1 + \theta_2)$ and $\sin\theta_1\cos\theta_2 + \sin\theta_2\cos\theta_1 = \sin(\theta_1 + \theta_2)$ to conclude that this is in fact $$r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)].$$