Prove that: 1+ sin 2x / sin^2 x = csc^2 x + 2cot x

1 + sin2x / sin^2 x = csc^2 x + 2cot
x

We will start from the right side.

We
know that csc x = 1/sinx and cotx = cosx/sinx

==> csc^2 x +
2cotx = (1/sin^2 x) + 2 cosx/sinx

We will find the common
denominator.

==> csc^2 x + 2cotx = (1+ 2sinxcosx)/sin^2
x

Now we know that 2sinx*cosx =
sin2x

==> csc^2 x + 2cotx = (1+ sin2x)/sin^2 x
........q.e.d

Then we proved that: csc^2 x + 2cotx =
(1+ sin2x) /sin^2 x