Question

Prove Identity: (1-cos2x/ tan x) = sin2x

Answer 1

I think the parentheses are in the wrong place. It should probably read:

(1-cos2x)/tan x = sin2x

Again, split everything into sin and cos, and don't forget the identities:
cos(2x)=cos²(x)-sin²(x)
sin(2x)=2sin(x)cos(x)

(1-cos(2x))/tan(x)
=(1-(cos²(x)-sin²(x))*(cos(x)/sin(x))
=(1-(1-2sin²(x))*cos(x)/sin(x)
=2sin²(x)*cos(x)/sin(x)
=2sin(x)cos(x)
=sin(2x)