Sin[[Pi]/12] // Simplify
(-1 + Sqrt[3])/(2 Sqrt[2])
How to rationalize the denominator get the result as following:
(Sqrt[6] - Sqrt[2])/4
Try Sin[\[Pi]/12] // FullSimplify which has a rational denominator but not the same form as you desired: $\frac{\sqrt{2-\sqrt{3}}}{2}$
Sin[\[Pi]/12] // FullSimplify
Here is a way:
Sin[\[Pi]/12] // Apart // Together
Adding // Simplify to what you wrote gets for form that the OP desired.
// Simplify
