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