Proving trig identities

1/(1-cos x) +1/(1+cos x)=2 csc^2 x

After getting the common denominator-->
2/(1-cos^2 x)

then,
2/sin^2 x

Now I know that 1/sin^2 x= csc^2 x, so

2/sin^2 x=1/2 csc^2 x

But I need 2 csc^2 x.....
What do I need to do?

Thanks in advance.

............
then,
2/sin^2 x

Now I know that 1/sin^2 x= csc^2 x, so
2/sin^2 x=1/2 csc^2 x
NO...! Why are you inverting the 2 along with sin?
2/sin^2 x= 2 (1/sin^2 x) = 2 csc^2 x

...............

omg. I don't know why I did that! I guess my brain function is ailing after doing 55 of these.

Thanks so much!