Solve sin x - cos x = 2 cos(2x):

sin x - cos x = 2 (cos^2 x - sin^2 x)

=> sin x - cos x = 2 (cos x - sin x)(cos x + sin x)

=> sin x - cos x = 0 or 1 = -2 (cos x + sin x)

=> tan x = 1 or cos x + sin x = -1/2.

To solve the second equation, note that cos x + sin x = sqrt{2} cos (x + pi/4).