Divide by 6
[tex]cos x - sin x = \frac{\sqrt{6}}{2}[/tex]
Square both sides
[tex]cos^2 x -2 sin x cos x + sin^2 x = \frac{3}{2}[/tex]
Simplify Left side using Trig identities: (Pythagorean, Double Angle)
[tex]1 - sin (2x) = \frac{3}{2}[/tex]
Solve for x:
[tex]sin (2x) = -\frac{1}{2} \\ \\ 2x = \frac{7 \pi}{6}, \frac{11 \pi}{6} \\ \\ x = \frac{7 \pi}{12}, \frac{11 \pi}{12}[/tex]
