$\underset{n\to \infty }{Lt}{\left(\sin \left(\frac{\pi }{2n}\right)\sin \frac{2\pi }{2n}\mathrm{........}\sin \frac{(n-1)\pi }{n}\right)}^{\frac{1}{n}}$

$y=\underset{n\to \alpha }{Lt}{\left(\sin \frac{\pi }{2n}\sin \frac{2\pi }{2n}\mathrm{..........}\sin \frac{2(n-1)\pi }{2n}\right)}^{1/n}$

$\log y=\underset{n\to \infty }{Lt}\frac{1}{n}\sum _{r=1}^{2(n-1)}\log \left(\frac{r\pi }{2n}\right)=\underset{0}{\overset{2}{\int }}\mathrm{logsin}\left(\frac{\pi x}{2}\right)dx$

$\Rightarrow y=\frac{1}{4}$