Question

The value of $\cot \frac{\pi }{20}\cot \frac{3\pi }{20}\cot \frac{5\pi }{20}\cot \frac{7\pi }{20}\cot \frac{9\pi }{20}$ is

Answer 1

Given
$\cot \frac{\pi }{20}\cot \frac{3\pi }{20}\cot \frac{5\pi }{20}\cot \frac{7\pi }{20}\cot \frac{9\pi }{20}=\left(\cot \frac{\pi }{20}\cot \frac{9\pi }{20}\right)\cot \frac{\pi }{4}\left(\cot \frac{3\pi }{20}\cot \frac{7\pi }{20}\right)=\left[\cot \frac{\pi }{20}\cot (\frac{\pi }{2}-\frac{\pi }{20})\right]\left[\cot \frac{3\pi }{20}\cot (\frac{\pi }{2}-\frac{3\pi }{20})\right]=\left(\cot \frac{\pi }{20}\tan \frac{\pi }{20}\right)\left(\cot \frac{3\pi }{20}\tan \frac{3\pi }{20}\right)=1$