Official solution
\[2\sin12^\circ-\sin72^\circ = \sin12^\circ + (\sin12^\circ - \sin72^\circ) = \sin12^\circ - \sin48^\circ\ (\text{using }\sin72^\circ=\cos18^\circ,\text{ etc.})\]
\[\sin12^\circ - \sin48^\circ = -2\cos30^\circ\sin18^\circ = -2\cdot\frac{\sqrt{3}}2\cdot\frac{\sqrt{5}-1}4 = \frac{\sqrt{3}(1-\sqrt{5})}4\]