Proof of the Addition Identies

[Bogrune]

So throughout this entire semester in my Precalculus class my instructor told us that for solving trigonometric formulas, we will only learn about using these three formulas:
sin²θ + cos²θ = 1
sin(A+B) = sinAcosB + cosAsinB

and

cos(A+B) = cosAcosB - sinAsinB

For about two months we've studied trigonometry using almost nothing but these three formulas, and I understood how to use them well. I understood proof of the sin²θ + cos²θ = 1 formula, since the formula for the unit circle is x² + y² = 1, but I've never understood the proof for the addition identities. Just out of curiosity, can anyone show me proof of both addition identities?

 

[dextercioby]

Please, see the mathworld site: http://mathworld.wolfram.com/TrigonometricAdditionFormulas.html and the wiki page http://en.wikibooks.org/wiki/Trigonometry/Addition_Formula_for_Sines

 

[Bogrune]

Huh... and to think that the last place that I was going to look for proof was Wikipedia since it always explains even the most simple math-related topic as if it was Complex Analysis. Thanks for the help though!