问题及解答

\[
\frac{\tan 12^{\circ}-\sqrt{3}}{(4\cos^2 12^{\circ}-2)\sin 12^{\circ}}
\]

\[
\begin{split}
\frac{\tan 12^{\circ}-\sqrt{3}}{(4\cos^2 12^{\circ}-2)\sin 12^{\circ}}&=\frac{\tan 12^{\circ}-\sqrt{3}}{(2\cos^2 12^{\circ}-1)2\sin 12^{\circ}}\\
&=\frac{\tan 12^{\circ}-\sqrt{3}}{\cos 24^{\circ}\cdot 2\sin 12^{\circ}}\\
&=\frac{\tan 12^{\circ}\cdot\cos 12^{\circ}-\sqrt{3}\cos 12^{\circ}}{\cos 24^{\circ}\cdot 2\sin 12^{\circ}\cdot\cos 12^{\circ}}\\
&=\frac{\sin 12^{\circ}-\sqrt{3}\cos 12^{\circ}}{\cos 24^{\circ}\cdot\sin 24^{\circ}}\\
&=\frac{2\bigl(\sin 12^{\circ}\cdot\frac{1}{2}-\cos 12^{\circ}\cdot\frac{\sqrt{3}}{2}\bigr)}{\frac{1}{2}\sin 48^{\circ}}\\
&=4\cdot\frac{\sin(12^{\circ}-60^{\circ})}{\sin 48^{\circ}}\\
&=4\cdot\frac{-\sin 48^{\circ}}{\sin 48^{\circ}}=-4.
\end{split}
\]