Example 1. Find the value of $\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}$.
Considering $\sin \alpha \cos \alpha=\frac{1}{2} \sin 2 \alpha$, we can use its symmetric (dual) form to assist in solving: Let $x=\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}, y=\cos 10^{\circ} \cos 30^{\circ} \cos 50^{\circ} \cos 70^{\circ}$, then
$$
\begin{aligned}
x y & =\frac{1}{16} \sin 20^{\circ} \sin 60^{\circ} \sin 100^{\circ} \sin 140^{\circ} \\
& =\frac{1}{16} \cos 70^{\circ} \cos 30^{\circ} \cos 10^{\circ} \cos 50^{\circ} \\
& =\frac{1}{16} y . \quad \text { (continued on page 39) }
\end{aligned}
$$
(continued on page 39)
Since $y \neq 0$,
$$
x=\sin 10^{\circ} \sin 30^{\circ} \sin 50^{\circ} \sin 70^{\circ}=\frac{1}{16} .
$$