sin12^osin24^osin48^osin84^o =

$\begin{array}{l}\sin {12}^{\circ }\sin {24}^{\circ }\sin {48}^{\circ }\sin {84}^{\circ }\\ =\frac{\sin {12}^{\circ }\sin {24}^{\circ }\sin \left(60-{24}^{\circ }\right)\sin (60+24)}{\sin (60+12)\sin (60-12)}\\ =\frac{\frac{1}{4}\mathrm{Xsin}{72}^{\circ }\sin 36}{\sin {36}^{\circ }\mathrm{X}\frac{1}{4}\mathrm{Xsin}{72}^{\circ }\mathrm{Xsin}{72}^{\circ }}=\frac{1}{16}\\ \cos {20}^{\circ }\cdot \cos {40}^{\circ }\cdot \cos {60}^{\circ }\cdot \cos {80}^{\circ }\\ =\cos {20}^{\circ }\cos \left(60+{20}^{\circ }\right)\cos \left(60-{20}^{\circ }\right)\cos {60}^{\circ }\\ =\frac{1}{4}\cos {60}^{\circ }\cdot \cos {60}^{\circ }\end{array}$
=1/16