How to prove?

The answer is as follows:

cos(2x) = cos(x)^2 - sin(x)^2
sin(2x) = 2sin(x)cos(x)
sin(t)^2 + cos(t)^2 = 1 for all values of t
cos(x/2) = sqrt((1/2) * (1 + cos(x)))
sin(x/2) = sqrt((1/2) * (1 - cos(x)))


(cos(2x) + 1) / sin(2x) =>
(cos(x)^2 - sin(x)^2 + 1) / (2sin(x)cos(x)) =>
(cos(x)^2 + cos(x)^2) / (2sin(x)cos(x)) =>
2cos(x)^2 / (2sin(x)cos(x)) =>
cos(x)/sin(x) =>
cos(x)/sin(x)  *  1 =>
(cos(x)/sin(x)) * (sin(x)/sin(x)) =>
sin(x)cos(x) / sin(x)^2 =>
sin(x)cos(x) / sin(2x/2)^2 =>
sin(x)cos(x) / ((1/2) * (1 - cos(2x))) =>
2 * sin(x)cos(x) / (1 - cos(2x)) =>
sin(2x) / (1 - cos(2x))