Question

Tan 5°× tan 25°× tan 30°×tan 65°× tan 85°

Answer 1

Use the identity
$\tan( \alpha ) \times \tan(60 - \alpha ) \times \tan(60 + \alpha ) \\ = \tan(3 \alpha )$
You can see that 65=60+5, and 85=60+25.
We also use the identity
$\tan( \alpha ) = \cot(90 - \alpha )$
at the end.
So
$\frac{ \tan(5) \tan(55) \tan(65) }{ \tan(55) } \times \frac{ \tan(25) \tan(35) \tan(85) }{ \tan(35) } \\ = \frac{ \tan(3 \times 5) }{ \tan(55) } \times \frac{ \tan(25 \times 3) }{ \tan(35) } \\ = \frac{ \tan(15) }{ \tan(55) } \times \frac{ \tan(75) }{ \tan(35) } \\ = \frac{ \tan(15) }{ \tan(55) } \times \frac{ \cot(15) }{ \cot(55) } \\ = 1$
The required answer
$1 \times \tan(30) \\ = \frac{1}{ \sqrt{3} }$