Example 1. Find the value of $\sin 20^{\circ} \sin 40^{\circ} \sin 60^{\circ} \sin 80^{\circ}$.
\begin{array}{l} \text { Original expression }=\sin 20^{\circ} \sin \left(60^{\circ}-20^{\circ}\right) \\ \cdot \sin \left(60^{\circ}+20^{\circ}\right) \sin 60^{\circ} \\ =\frac{1}{4} \sin ^{2} 60^{\circ}=\frac{3}{16} .\end{array}