Looks like the claim is
(2 tan(x) - sin(2x)) / (2 sin²(x)) = tan(x)
Recall the double angle identity for sine:
sin(2x) = 2 sin(x) cos(x)
as well as the definition of tangent:
tan(x) = sin(x) / cos(x)
Then on the left side, we can cancel factors of 2 sin(x) to get
(2 sin(x) / cos(x) - 2 sin(x) cos(x)) / (2 sin²(x))
= (1/cos(x) - cos(x)) / sin(x)
Multiply this by cos(x)/cos(x) :
(1/cos(x) - cos(x)) / sin(x) = (1 - cos²(x)) / (sin(x) cos(x))
Recall that for all x,
sin²(x) + cos²(x) = 1
so that
(1 - cos²(x)) / (sin(x) cos(x)) = sin²(x) / (sin(x) cos(x))
= sin(x) / cos(x)
= tan(x)
as required.
