$\sin {12}^0\sin {48}^0\sin {54}^0=$

$=\sin {12}^{\circ }\sin {48}^{\circ }\sin {54}^{\circ }$

$=\frac{1}{2}\left[2\sin {12}^{\circ }\cos {42}^{\circ }\right]\sin {54}^{\circ }$

$=\frac{1}{2}\left[\sin {54}^{\circ }-\sin {30}^{\circ }\right]\sin {54}^{\circ }$

$=\frac{1}{2}\left[{\sin }^2{54}^{\circ }-\frac{1}{2}\sin {54}^{\circ }\right]$

$=\frac{1}{4}\left[2{\sin }^2{54}^{\circ }-\sin {54}^{\circ }\right]$

$=\frac{1}{4}\left[2{\left(\frac{1+\sqrt{5}}{4}\right)}^2-\left(\frac{1+\sqrt{5}}{4}\right)\right]$

$=\frac{1}{4}\left[2\left(\frac{1+5+2\sqrt{5}}{16}\right)-\left(\frac{1+\sqrt{5}}{4}\right)\right]$

$=\frac{1}{4}\times \frac{1}{8}[6+2\sqrt{5}-2-2\sqrt{5}]$

$=\frac{1}{32}\times 4=\frac{1}{8}$