First, I put $t = \sin x - \cos x$,

 eq1 = (3 - Cos[4x]) ( 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] ∈ Integers && x == π/2 + 2 π C[1]) || (C[1] ∈ Integers && x == π + 2 π C[1])
This is my solution by hand, I put at https://math.stackexchange.com/questions/218381/how-to-solve-this-trigonometric-equation/218496#218496