Answer
$\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$ and $\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta$.
Explanation
Setting $\alpha = \beta$ immediately yields the double-angle identities $\sin(2\alpha) = 2\sin\alpha\cos\alpha$ and $\cos(2\alpha) = \cos^2\alpha - \sin^2\alpha$.