Question

tan 5° tan 25° tan 30° tan 65° tan 85° = ?
(a) 1
(b) $\sqrt{3}$
(c) $\frac{1}{\sqrt{3}}$
(d) $\frac{1}{2}$

Answer 1

$$\begin{gathered} \tan 5°\tan 25°\tan 30°\tan 65°\tan 85° \\ =\tan \left(90°-85°\right)\tan 25°\tan 30°\tan \left(90°-25°\right)\tan 85° \\ =\cot 85°\tan 25°\tan 30°\cot 25°\tan 85°\left(\because \tan \left(90°-\theta \right)=\cot \theta \right) \\ =\left(\frac{1}{\tan 85°}\tan 85°\right)\left(\tan 25°\frac{1}{\tan 25°}\right)\tan 30°\left(\because \cot \theta =\frac{1}{\tan \theta }\right) \\ =\tan 30° \\ =\frac{1}{\sqrt{3}} \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{option}\mathrm{is}\left(\mathrm{c}\right). \end{gathered}$$