$tan(\frac{\pi}{12})=\frac{sin(\frac{\pi}{12})}{cos(\frac{\pi}{12})}$\\
$=\frac{\frac{\sqrt{2}-\sqrt{6}}{4}}{\frac{\sqrt{2}+\sqrt{6}}{4}}$\\
$=\frac{\sqrt{2}-\sqrt{6}}{4} \times {\frac{4}{\sqrt{2}+\sqrt{6}}}$\\
$=\frac{\sqrt{2}-\sqrt{6}}{\sqrt{2}+\sqrt{6}}$\\
$=\frac{(\sqrt{2}-\sqrt{6})\times(\sqrt{2}-\sqrt{6})}{(\sqrt{2}+\sqrt{6})\times(\sqrt{2}-\sqrt{6})}$\\
$=\frac{(\sqrt{2}-\sqrt{6})^2}{\sqrt{2}^2-\sqrt{6}^2}$\\
$=\frac{(\sqrt{2}-\sqrt{6})^2}{2-6}$\\
$=\frac{2-4\sqrt{3}+6}{-4}$\\
$=\frac{4\sqrt{3}+8}{-4}$\\
$=-\sqrt{3}-2$