We use the trig identity for the difference of two cosines:
cos(2x)-cos(3x)=2sin(5x/2)sin(x/2).
Also, cos(4x)-1=1-2sin²(2x)-1=-2sin²(2x).
As x→0, sin(5x/2)→5x/2, sin(x/2)→x/2, sin(2x)→2x,
therefore (cos(2x)-cos(3x))/cos(4x-1)→(2)(5x/2)(x/2)/(-(2)(4x²))→
(10x²/4)/(-8x²)=-5/16 QED.