Without using an electronic calculator?

Another approach is to prove a useful identity and then use it.

$\begin{align*}4\cos(\theta)\cos(\theta + 60^\circ)\cos(\theta - 60^\circ) &= 2\cos(\theta)(\cos(2\theta) + \cos(120^\circ)) \\

&= 2\cos(\theta)(2\cos^2(\theta) - 3/2) \\

&= 4\cos^3(\theta) - 3\cos(\theta) \\

&= \cos(3\theta) \end{align*}$

For $\theta = 20^\circ$, $4\cos(20^\circ)\cos(80^\circ)\cos(40^\circ) = \cos(60^\circ) = 1/2$,

and.multiplying both sides of that by $2\cot(10^\circ)$ gives the original equation to be verified.