Derivation of Sum and Difference Angles

$\displaystyle \sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$

$\displaystyle \sin(\alpha - \beta) = \sin\alpha\cos\beta - \cos\alpha\sin\beta$

$\displaystyle \cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$

$\displaystyle \cos(\alpha - \beta) = \cos\alpha\cos\beta + sin\alpha\sin\beta$

and then there is $\displaystyle \tan$.

could someone tell me how these sum and difference angle formulas are derived?

i've tried working it out but no luck. you don't have to do all of them, just one will suffice.

thanks