$\sin \left( \frac{\pi }{2}+\alpha  \right)\sin (\pi +\alpha )-\sin (\pi -\alpha )\cos (-\alpha )$ می‌دانیم $\sin (\pi -\alpha )=\sin \alpha \,\,\And \,\,\sin (\pi +\alpha )=-\sin \alpha $ و $\sin \left( \frac{\pi }{2}+\alpha  \right)=\cos \alpha $ بنابراین: $\begin{align}  & =(\cos \alpha )(-\sin \alpha )-(\sin \alpha )(\cos \alpha ) \\  & =-\sin \alpha \,\cos \alpha -\sin \alpha \,\cos \alpha  \\  & =-2\sin \alpha \,\cos \alpha =-\sin 2\alpha  \\ \end{align}$