Question

Find the values of $\tan \frac{\pi }{20}\tan \frac{3\pi }{20}\tan \frac{5\pi }{20}\tan \frac{7\pi }{20}\tan \frac{9\pi }{20}$

Answer 1

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