Prove that $\tan10 \cdot \tan20 \cdot \tan30 \cdot \ldots \cdot \tan80$ is a rational number.

Question

Prove that $\tan10 \cdot \tan20 \cdot \tan30 \cdot \ldots \cdot \tan80$ is a rational number.

I have no idea on where to start with this question, so any pointers would be appreciated.

Answer 1

(Noting that you're working with degrees rather than radians here)

First write each $tan(x)$ as $\frac{sin(x)}{cos(x)}$. Then use the fact that $sin(x)=cos(90-x)$ to pair up factors in the numerator and denominator.

Answer 2

Hint:

Group in pairs the tangents of complementary angles.

Answer 3

Note that $tan(10)*tan(80) = 1$, $tan(20)*tan(70) = 1$ et caetera $tan(k10)*tan(90-k10) = 1$.