The value of $\sum _{n=1}^{10}(\sin \frac{2n\pi }{11}-\cos \frac{2n\pi }{11})$ is

G.E. $=(\sin \frac{2\pi }{11}+\sin \frac{4\pi }{11}+\cdots +\sin \frac{20\pi }{11})-(\cos \frac{2\pi }{11}+\cos \frac{4\pi }{11}+\cdots +\cos \frac{20\pi }{11})$

$=\frac{\sin \pi \cdot \sin \frac{10\pi }{11}}{\sin \frac{\pi }{11}}-\frac{\cos \pi \cdot \sin \frac{10\pi }{11}}{\sin \frac{\pi }{11}}$

$=0+\frac{\sin (\pi -\frac{\pi }{11})}{\sin \frac{\pi }{11}}=1$.