13. (2004 National College Entrance Examination - Jiangsu Paper) Given: $0<\alpha<\frac{\pi}{2}, \tan \frac{\alpha}{2}+\cot \frac{\alpha}{2}=\frac{5}{2}$. Find the value of $\sin \left(\alpha-\frac{\pi}{3}\right)$.
13. $\frac{1}{10}(4-3 \sqrt{3})$ From $\tan \frac{\alpha}{2}+\cot \frac{\alpha}{2}=\frac{5}{2}$ we get $\sin \alpha=\frac{4}{5}$, so $\cos \alpha=\frac{3}{5}, \sin \left(\alpha-\frac{\pi}{3}\right)=\sin \alpha \cos \frac{\pi}{3}-$ $\cos \alpha \sin \frac{\pi}{3}=\frac{1}{10}(4-3 \sqrt{3})$.