We have to prove that (csc x)^2 = (2 sec 2x) / (sec 2x - 1)
(2 sec 2x) / (sec 2x - 1)
=> [2*(1/ cos 2x)] / [((1/ cos 2x) - 1)]
=> [2*(1/ cos 2x)] / [((1 - cos 2x)/ cos 2x)]
=> 2/ (1 - cos 2x)
=> 2 / ( 1- (1 - 2 (sin x)^2))
=> 2 / [ 1 - 1 + 2(sin x)^2]
=> 2 / 2 (sin x)^2
=> cosec x
This proves that the right hand side = left hand side
Therefore we prove that (csc x)^2 = (2 sec 2x) / (sec 2x - 1)