Prove that
2 sin(2x) = 4 sin(x)·cos(x)
sin(4x) = 2 sin(2x)·cos(2x) = 4sin(x)·cos(x)·cos(2x)
cos(2x) = cos(x)-sin(x)
→
Ok, with that solved we can now begin.
(2 sin(2x)-sin(4x))/(2sin(2x)+sin(4x)) =
(4sin(x)·cos(x)-4sin(x)·cos(x)·cos(2x))/(4(sin(x)·cos(x)+4sin(x)·cos(x)·cos(2x) =
(4sin(x)·cos(x)(1-cos(2x)))/(4sin(x)·cos(x)(1+cos(2x)) =
(1-cos(2x))/(1+cos(2x)) =
(1-(cos(x)-sin(x))/(1+cos(x)-sin(x)) =
(sin(x)+cos(x)-cos(x)+sin(x))/(sin(x)+cos(x)+cos(x)-sin(x) =
2sin(x)/2cos(x) = tan(x)