Question

Calculate without resorting to table.
$\tan {20}^{\circ }\tan {40}^{\circ }\tan {80}^{\circ }$

Answer 1

$\tan 30°\tan 40°\tan 80°$

We know that $\tan (60-A)\tan A\tan (60+A)=\tan 3A$

We put $A=40,$ we get

$\tan 20°\tan 40°\tan 100°=\tan 120°$

$\tan 20°\tan 40°\tan (180°-80°)=\tan 120°$

$\tan 20°\tan 40°\tan 80°=-\tan 120°=\tan (180°-60°)$

$\tan 20°\tan 40°\tan 80°=\tan 60°=\sqrt{3}$