The value of \(\sin 10^{\circ} \cdot \sin 30^{\circ} \cdot \sin 50^{\circ} \cdot \sin 70^{\circ}\) is

\(\begin{aligned} \sin 10^{\circ} \cdot & \sin 30^{\circ} \cdot \sin 50^{\circ} \cdot \sin 70^{\circ} \\ &=\frac{1}{2} \cdot \sin 10^{\circ} \cdot \frac{1}{2}\left(2 \sin 70^{\circ} \cdot \sin 50^{\circ}\right) \\ &=\frac{1}{2} \sin 10^{\circ} \cdot \frac{1}{2}\left\{\cos \left(70^{\circ}-50^{\circ}\right)\right.\\ &=\frac{1}{2} \sin 10^{\circ} \cdot \frac{1}{2}\left\{\cos \left(70^{\circ}+50^{\circ}\right)\right\} \\ &=\frac{1}{2} \sin 10^{\circ} \cdot \frac{1}{2}\left(\cos 10^{\circ}+\frac{1}{2}\right) \\ &=\frac{1}{4} \sin 10^{\circ} \cdot \cos 20^{\circ}+\frac{1}{8} \sin 10^{\circ} \\ &=\frac{1}{4} \cdot \frac{1}{2}\left(\sin 30^{\circ}-\sin 10^{\circ}\right)+\frac{1}{8} \cdot \sin 10^{\circ} \\ &=\frac{1}{8} \cdot \sin 30^{\circ}-\frac{1}{8} \sin 10^{\circ}+1 / 8 \sin 10^{\circ} \\ &=\frac{1}{8} \cdot \frac{1}{2}-0=\frac{1}{16} \end{aligned}\)