Explanation

(c)
$\sin{20{^\circ}\sin{40{^\circ}\sin{60{^\circ}\sin{80{^\circ}}}}}$
$= \frac{1}{2}\sin{20{^\circ}\sin{60{^\circ}(2\sin{40{^\circ}\sin{80{^\circ}})}}}$
$= \frac{1}{2}\sin{20{^\circ}\sin{60{^\circ}(\cos{40{^\circ}} - \cos{120{^\circ})}}}$
$= \frac{1}{2} \bullet \frac{\sqrt{3}}{2}\sin{20{^\circ}\left( 1 - 2\sin^{2}{20{^\circ} + \frac{1}{2}} \right)}$
$= \frac{\sqrt{3}}{4}\sin{20{^\circ}\left( \frac{3}{2} - 2\sin^{2}{20{^\circ}} \right)}$
$= \frac{\sqrt{3}}{8}(3\sin{20{^\circ} - 4\sin^{3}{20{^\circ}}})$
$= \frac{\sqrt{3}}{8}\sin{60{^\circ} = \frac{\sqrt{3}}{8} \bullet \frac{\sqrt{3}}{2} = \frac{3}{16}}$