Prove that 2cos2x/sin2x=cotx-tanx, show every step on how to solve it

2cos2x/sin2x=cotx-tanx

2cos2x/sin2x = 2 (cos²x-sin²x) / (2sinx*cosx) = (2 cos²x-2sin²x) / (2sinx*cosx) =

= 2cos²x / (2sinx*cosx) - 2sin²x) / (2sinx*cosx) = cos x / sin x - sinx/cos x =

= cot x-tan x

Left side is equal to right side, so 2cos2x/sin2x=cotx-tanx is true.