Question

tan 5° tan 25° tan 30° tan 65° tan 85° = ?
(a) $\sqrt{3}$
(b) $\frac{1}{\sqrt{3}}$
(c) 1
(d) none of these

Answer 1

$$\begin{gathered} \text{The correct option is}\left(\text{b}\right). \\ \text{We have:} \\ {\text{tan5}}^0\tan {25}^0\tan {30}^0.\tan {65}^0\text{t}an{85}^0 \\ ={\text{tan5}}^0\tan {25}^0\tan {30}^0\tan ({90}^0-{25}^0)\text{t}an({90}^0-5^0) \\ =\tan 5^0\tan {25}^0\times \frac{1}{\sqrt{3}}\times \cot {25}^0\cot 5^0[\because \tan ({90}^0-\theta )=\cot \theta {\text{and tan30}}^0=\frac{1}{\sqrt{3}}] \\ =\frac{1}{\sqrt{3}} \end{gathered}$$