Using Solve you should incluede InverseFunctions -> True :
s = Solve[(3 - Cos[4*x])*(Sin[x] - Cos[x]) == 2, x, InverseFunctions -> True]
with Reduce you needn't use any options :
Reduce[(3 - Cos[4*x])*(Sin[x] - Cos[x]) == 2, x]
You can check that at least three solutions are rational multiples of Pi and for the other ones Mathematica cannot decide whether they are transcendental or algebraic numbers :
Element[#, Algebraics] & /@ s[[All, 1, 2]]