Question

The value of the equation $\tan \left(\frac{\pi }{12}\right)\cdot \tan \left(\frac{\pi }{16}\right)\cdot \tan \left(\frac{5\pi }{12}\right)\cdot \tan \left(\frac{7\pi }{16}\right)$ is

Answer 1

$\tan \left(\frac{\pi }{12}\right)\tan \left(\frac{\pi }{16}\right)\tan \left(\frac{5\pi }{12}\right)\tan \left(\frac{7\pi }{16}\right)$

$=$$\tan \left(\frac{\pi }{12}\right)\tan \left(\frac{\pi }{16}\right)\tan \left(\frac{5\pi }{12}\right)\tan \left(\frac{7\pi }{16}\right)$.

We know that $\frac{\pi }{12}=\frac{180}{12}={15}^{\circ }$

And $\frac{5\pi }{12}=5\times 15={75}^{\circ }$

$=\tan {15}^{\circ }\tan {75}^{\circ }\tan \frac{\pi }{16}\tan (\frac{\pi }{2}-\frac{\pi }{16})$

$=\tan {15}^{\circ }\tan ({90}^{\circ }-{15}^{\circ })\tan \frac{\pi }{16}\tan (\frac{\pi }{2}-\frac{\pi }{16})$

$=\tan {15}^{\circ }\cot {15}^{\circ }\tan \frac{\pi }{16}\cot \frac{\pi }{16}$

$=1\times 1=1$