Question

$\tan {20}^{\circ }.\tan {40}^{\circ }.\tan {60}^{\circ }.\tan {80}^{\circ }=k$. Find the value of $k$.

Answer 1

$\tan {20}^o\tan {40}^o\tan {60}^o\tan {80}^o=\tan ({60}^o-{20}^o)\tan \left({20}^o\right)\tan ({60}^o+{20}^o)\tan \left({60}^o\right)=\tan (3\times {20}^o)\tan {60}^o={\tan }^2{60}^o=3(\because \tan (60-\theta )\tan \theta \tan (60+\theta )=\tan 3\theta )$