Question

$\tan 5^{\circ }\tan {25}^{\circ }\tan {30}^{\circ }\tan {65}^{\circ }\tan {85}^{\circ }=$

• A. $1$
• B. $\frac{1}{2}$
• C. $\sqrt{3}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

The correct option is D $\frac{1}{\sqrt{3}}$

$\left(\tan 5^{\circ }\tan {85}^{\circ }\right)\left(\tan {25}^{\circ }\tan {65}^{\circ }\right)\tan {30}^{\circ }$
$=\left[\tan 5^{\circ }\tan \right({90}^{\circ }-5^{\circ }\left)\right]\left[\tan {25}^{\circ }\right(\tan {90}^{\circ }-{25}^{\circ }\left)\right]\times \tan {30}^{\circ }$
$=\tan 5^{\circ }\cot 5^{\circ }\tan {25}^{\circ }\cot {25}^{\circ }\times \frac{1}{\sqrt{3}}$
$=1\times 1\times \frac{1}{\sqrt{3}}=\frac{1}{\sqrt{3}}$.