Using complex numbers and exponential form perhaps help (at least algebraically) to digest these trigonometric addition formulas:
All we have to know is $\cos a+i\cdot\sin a=e^{ai}$ for any $a\in\Bbb R$, and that $i^2=-1$, and that $e^{x+y}=e^x\cdot e^y$ for any $x,y\in\Bbb C$.
Then calculate both sides of $e^{(a+b)i}=e^{ai}e^{bi}$.
If you prefer, instead, you can use the matrices of rotation:
$$R_a:=\pmatrix{\cos a&-\sin a\\ \sin a &\cos a}$$
and use matrix multiplication to verify the identities, knowing that
$$R_{a+b}=R_a\cdot R_b \ .$$