The Sum of Two Angles Formula

Given

$$e^{+\imath \theta} = \cos \theta + \imath \sin \theta\ \ \;\ \ e^{+\imath \phi} = \cos \phi + \imath \sin \phi$$

the product of these two is

$$e^{+\imath \theta} e^{+\imath \phi} = (\cos \theta + \imath... ... \left ( \cos \theta \sin \phi + \sin\theta \cos \phi \right )$$

but

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

implying (using the fact the the Real part of the l.h.s. equals the Real part of the r.h.s.,

$$\cos(\theta+\phi) = \cos \theta \cos \phi - \sin\theta \sin \phi$$

and likewise for the Imaginary part, i.e.,

$$\sin (\theta + \phi)= \cos \theta \sin \phi + \sin\theta \cos \phi$$