Question

Evaluate tan 5 tan 25 tan 30 tan 65 tan 85.

Answer 1

$$\begin{gathered} \tan 5°.\tan 25°.\tan 30°.\tan 65°.\tan 85° \\ =\tan 5°.\tan 85°.\tan 25°.\tan 65°.\tan 30° \\ =\tan 5°.\tan \left(90°-5°\right).\tan 25°.\tan \left(90°-25°\right).\frac{1}{\sqrt{3}} \\ =\tan 5°\times \cot 5°\times \tan 25°\times \cot 25°\times \frac{1}{\sqrt{3}} \\ =\tan 5°\times \frac{1}{\tan 5°}\times \tan 25°\times \frac{1}{\tan 25°}\times \frac{1}{\sqrt{3}} \\ =\frac{1}{\sqrt{3}} \end{gathered}$$