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)**