Derive Sin(x+y) & Cos(x+y) Using Euler's Formula

[Pseudo Statistic]

Can anyone tell me how to derive the sin(x+y) and cos(x+y) expansions? The ones that are like cos x sin y or sin y cos x + other stuff?
Preferrably, could this be derived with Euler's formula alone? Or something not too geometric? (All those OAs and OBs and XBs and XYs on geometric diagrams confuse me too much to follow)
Thank you.

 

[mathman]

You could use Euler's formula. It is tedious, but straightforward.

 

[StatusX]

Another way is to use the 2X2 rotation matrices R($\theta$), which have R(x)R(y)=R(x+y). This is equivalent to using Euler's formula, only you're working in R^2 instead of the complex numbers.

 

[VietDao29]

There's a short proof at wikipedia, you can view it at the end of this page.
It is, however, not a completed proof, but you can get some ideas about proving it. :)