From Förberedande kurs i matematik 1

Using the formula for double angles on \displaystyle \sin 160^{\circ} gives

\displaystyle \sin 160^{\circ} = 2\cos 80^{\circ}\sin 80^{\circ}\,\textrm{.}

On the right-hand side, we see that the factor \displaystyle \cos 80^{\circ} has appeared, and if we use the formula for double angles on the second factor (\displaystyle \sin 80^{\circ}),

\displaystyle 2\cos 80^{\circ}\sin 80^{\circ} = 2\cos 80^{\circ}\cdot 2\cos 40^{\circ}\sin 40^{\circ}\,,

we obtain a further factor \displaystyle \cos 40^{\circ}. A final application of the formula for double angles on \displaystyle \sin 40^{\circ } gives us all three cosine factors,

\displaystyle 2\cos 80^{\circ}\cdot 2\cos 40^{\circ}\cdot\sin 40^{\circ} = 2\cos 80^{\circ}\cdot 2\cos 40^{\circ}\cdot 2\cos 20^{\circ}\sin 20^{\circ}\,\textrm{·}

We have thus succeeded in showing that

\displaystyle \sin 160^{\circ} = 8\cos 80^{\circ}\cdot \cos 40^{\circ}\cdot \cos 20^{\circ}\cdot\sin 20^{\circ}

which can also be written as

\displaystyle \cos 80^{\circ}\cdot\cos 40^{\circ}\cdot \cos 20^{\circ} = \frac{\sin 160^{\circ}}{8\sin 20^{\circ}}\,\textrm{.}

If we draw the unit circle, we see that \displaystyle 160^{\circ} makes an angle of \displaystyle 20^{\circ} with the negative x-axis, and therefore the angles \displaystyle 20^{\circ} and \displaystyle 160^{\circ} have the same y-coordinate in the unit circle, i.e.

\displaystyle \sin 20^{\circ} = \sin 160^{\circ}\,\textrm{.}

This shows that

\displaystyle \cos 80^{\circ} \cos 40^{\circ} \cos 20^{\circ} = \frac{\sin 160^{\circ}}{8\sin 20^{\circ}} = \frac{1}{8}\,\textrm{.}