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