First, I put $t = \sin x - \cos x$,
eq1 = (3 - Cos[4*x])*(Sin[x] - Cos[x]) - 2 == 0; eq2 = t == Sin[x] - Cos[x] Eliminate[TrigExpand[{eq1, eq2}], x] I receive
2 t - 2 t^3 + t^5 == 1
And then, I solve
Solve[2 t - 2 t^3 + t^5 == 1, Reals] finally
Reduce[-Cos[x] + Sin[x] == 1, x, Reals] (C[1] \[Element] Integers && x == \[Pi]/2 + 2 \[Pi] C[1]) || (C[1] \[Element] Integers && x == \[Pi] + 2 \[Pi] C[1]) This is my solution by hand, I put at http://math.stackexchange.com/questions/218381/how-to-solve-this-trigonometric-equation/218496#218496