Explanation

(d)
$\sin{36{^\circ}\sin{72{^\circ}\sin{108{^\circ}\sin{144{^\circ}}}}}$
$= \sin^{2}{36{^\circ}\sin^{2}{72{^\circ}}}$
$= \frac{1}{4}\left\lbrack \left( 2\sin^{2}{36{^\circ}} \right)(2\sin^{2}{72{^\circ})} \right\rbrack$
$= \frac{1}{4}\left\lbrack \left( 1 - \cos 72{^\circ} \right)(1 - \cos 144{^\circ}) \right\rbrack$
$= \frac{1}{4}\left\lbrack \left( 1 - \sin 18{^\circ} \right)(1 + \cos 36{^\circ}) \right\rbrack$
$= \frac{1}{4}\left\lbrack \left( 1 - \frac{\sqrt{5} - 1}{4} \right)\left( 1 + \frac{\sqrt{5} + 1}{4} \right) \right\rbrack$
$= \frac{1}{4}\left\lbrack 1 + \left( \frac{\sqrt{5} + 1}{4} \right) - \left( \frac{\sqrt{5} - 1}{4} \right) - \left( \frac{4}{16} \right) \right\rbrack$
$= \frac{1}{4}\left\lbrack 1 + \frac{1}{2} - \frac{1}{4} \right\rbrack = \frac{5}{16}$