How to Prove the Trigonometric Identities for Sin and Cos of A+B?

[basty]

Why

$\sin (A+B) = \sin A \cos B + \cos A \sin B$

and

$\cos (A+B) = \cos A \cos B - \sin A \sin B$

??

 

[Orodruin]

The proof of this should be available in any basic textbook on trigonometry or by a simple search in a search engine. I suggest starting here: http://en.wikipedia.org/wiki/Proofs_of_trigonometric_identities
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