Complex exponential function identity proof

Question

I see the following identity in my book however they don't prove it so I am wondering how to prove that $e^{i \theta_1+i\theta_2}=e^{i\theta_1}e^{i\theta_2}$?

The definition is:

$$e^{i\theta} = \cos(\theta)+i\sin(\theta)$$

Answer 1

Hint $$e^{i\theta_1}e^{i\theta_2}=\bigl(\cos(\theta_1)+i\sin(\theta_1) \bigr) \bigl(\cos(\theta_2)+i\sin(\theta_2)\bigr)$$

Just open the brackets.