Use \(\sin(u\pm v)=\sin u\cos v\pm\cos u\sin v\):
\( \sin(\alpha+\beta)+\sin(\alpha-\beta)\) \(=(\sin\alpha\cos\beta+\cos\alpha\sin\beta)\) \(+(\sin\alpha\cos\beta-\cos\alpha\sin\beta)\) \(=2\sin\alpha\cos\beta. \)