Simplify
\(\sin(\alpha-\beta)\cos\alpha-\cos(\alpha-\beta)\sin\alpha\).
Solution
Recognize the sine–difference pattern
\(\sin U\cos V-\cos U\sin V=\sin(U-V)\) with \(U=\alpha-\beta\), \(V=\alpha\):
\(
\sin(\alpha-\beta)\cos\alpha-\cos(\alpha-\beta)\sin\alpha\)
\(=\sin\big((\alpha-\beta)-\alpha\big)\)
\(=\sin(-\beta)=-\sin\beta.
\)
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