[Trigonometry] Sum-Difference Identity

Hi r/learnmath, I'm stuck trying to figure out how I got the answer wrong and I need a second set of eyes! The question is:

[;\begin{matrix} &\cos(\frac{5\pi}{12}) &= &\cos(\frac{\pi}{4}+\frac{\pi}{6})\ \ & &= &\cos(\frac{\pi}{4})\cdot\cos(\frac{\pi}{6})-\sin(\frac{\pi}{4})\cdot\sin(\frac{\pi}{6})\ \ & &= &\frac{\sqrt{2}}{2}\cdot\frac{1}{2}-\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{3}}{2}\ \ & &= &\frac{\sqrt{2}}{4}-\frac{\sqrt{6}}{4}\ \ & &= &\frac{\sqrt{2}-\sqrt{6}}{4}\ \ &The \ correct \ answer \ is:\ \ & &= &\frac{\sqrt{6}-\sqrt{2}}{4} \end{matrix};]