Question

The value of $\tan \left(\frac{\pi }{20}\right)\tan \left(\frac{3\pi }{20}\right)\tan \left(\frac{5\pi }{20}\right)\tan \left(\frac{7\pi }{20}\right)\tan \left(\frac{9\pi }{20}\right)$ is

• A. $0$
• B. $1$
• C. $-1$
• D. $\infty$

Answer 1

The correct option is B $1$

$\tan \left(\frac{\pi }{20}\right)\tan \left(\frac{3\pi }{20}\right)\tan \left(\frac{5\pi }{20}\right)\tan \left(\frac{7\pi }{20}\right)\tan \left(\frac{9\pi }{20}\right)$

$=\tan \left(\frac{\pi }{20}\right)\tan \left(\frac{3\pi }{20}\right)\tan \left(\frac{5\pi }{20}\right)\tan (\frac{\pi }{2}-\frac{3\pi }{20})\tan (\frac{\pi }{2}-\frac{\pi }{20})$

$=\tan \left(\frac{\pi }{20}\right)\tan \left(\frac{3\pi }{20}\right)\tan \left(\frac{\pi }{4}\right)\cot \left(\frac{3\pi }{20}\right)\cot \left(\frac{\pi }{20}\right)$

$=1$