\(\operatorname{Sin} 20^{\circ} \operatorname{Sin} 30^{\circ} \operatorname{Sin} 40^{\circ} \operatorname{Sin} 80^{\circ}\)
\(=\sin 20^{\circ} \times \frac{1}{2} \times \sin 40^{\circ} \sin 80^{\circ} \)
\(=\frac{1}{2} \times \sin 20^{\circ} \sin 40^{\circ} \sin 80^{\circ} \)
\(=\frac{1}{2} \times \sin 20^{\circ} \sin \left(60^{\circ}-20^{\circ}\right) \sin \left(60^{\circ}+20^{\circ}\right) \)
\(=\frac{1}{2} \times \frac{1}{4} \sin \left(3 \times 20^{\circ}\right) \quad\left[\because \sin x \sin \left(60^{\circ} -x\right) \sin \left(60^{\circ}+x\right)=\frac{1}{4} \sin 3x\right] \)
\(=\frac{1}{8} \sin 60^{\circ} \)
\(=\frac{1}{8} \times \frac{\sqrt{3}}{2} \)
\(=\frac{\sqrt{3}}{16} .\)
