Prove the following identity: (csc^2) x = (2 sec 2x) / (sec 2x - 1)

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)