Question

Evaluate : $\frac{2}{\sqrt{3}}tan{5}^{\circ }.tan{15}^{\circ }tan60.tan{75}^{\circ }.tan{85}^{\circ }$

Answer 1

$\frac{2}{\sqrt{3}}\tan 5^{\circ }\tan {15}^{\circ }\tan {60}^{\circ }\tan {75}^{\circ }\tan {85}^{\circ }=\frac{2}{\sqrt{3}}\tan 5^{\circ }\tan {15}^{\circ }\tan {60}^{\circ }\tan ({90}^{\circ }-{15}^{\circ })\tan ({90}^{\circ }-5^{\circ })$

$=\frac{2}{\sqrt{3}}\left(\tan 5^{\circ }\cot 5^{\circ }\right)\left(\tan {15}^{\circ }\cot {15}^{\circ }\right)\left(\tan {60}^{\circ }\right)$

$=\frac{2}{\sqrt{3}}\left(1\right)\left(1\right)\left(\sqrt{3}\right)=2$