reverse chain rule question

When doing ∫ cosec^2 2x cot 2x dx, I get -(1/2)(cot ^2 2x) + c, but I was curious and used cosec^2 2x with reverse chain rule and I cant seem to get the same value or anything equivalent. Is it wrong to use cosec^2x in the reverse chain rule or have i just gotten the wrong answer

Seems ok to me, but youre out by a (1/2)? If youre unsure about your working maybe post your explanation for reverse chain or explicitly do a substitution if youve covered that. But you can express the ans in terms of cosec(2x) using the pythagorean identity / do the reverse chain differently.

I get this, even with the identity I cant see them being equivalent and when I differentiate my solution and dont end up with what I started with

You had cot^2(2x) in the op and using the pythagorean identity and ignoring the +1 as thats another pesky constant thats cosec^2(2x). So diff either/both of those and see what you get, then think about how to reverse it, so which term(s) is the derviative and which is the square youre looking for.

but its cosec^3 2x not ^2 so the identity would uphold

You had cot^2(2x) in the op and using the pythagorean identity, and ignoring the +1 as thats another pesky constant, thats cosec^2(2x). So diff either/both of those and see what you get, then think about how to reverse it/them, so which term(s) is/are the derviative and which is the square youre looking for.
Not sure where the cube part for y comes from in your working (its not right).

Why is it a cube?

Dont you have to increase the power by 1, its what ive done for every question that ive answered and its worked

Sounds a solid reason to me :-).
So the derivative of
f^2(x)
is
2f'(x)f(x)
So if it was cot^2, whats the derivative of cot?

Derivative of cot ^2 x
(cot x )^2 --> 2cot x sec^2x = 2cotx sec^2x which that works with the question. Thats where my confusion lies because it doesnt work when you do the same thing with the cosec

Sure so (forgetting about the double angles / constants)
cot^2 <-> 2 (cosec^2) cot
so you pick sec^2 as the dervative of cot and get cot^2 (+c) when you do the revese chain/anti derivative.
So same thing with cosec^2 is ....
It must be the same thing as the pythagorean identity says
cot^2 = cosec^2 (+c)

-cosec ^2 is the derivative of cot not sec^2 though

Sure so (forgetting about the double angles / constants)
cot^2 <-> 2 (cosec^2) cot
so you pick cosec^2 as the dervative of cot and get cot^2 (+c) when you do the revese chain/anti derivative.
So same thing with cosec^2 is ....
It must be the same thing as the pythagorean identity says
cot^2 = cosec^2 (+c)

Im not really sure what you are saying, so are you saying that - cosec ^3 / 6 + c is the same as -1/4 cot ^2 +c

No. Have you tried differnetiating cosec^2 and thefrefore spotting what the reverse chain is?
If you think about it, the deriviative of the square cot^2 is a cubic in trig terms so cosec^2 cot, so differntiating increases the order/integration decreases the order.

Yes but using the reverse chain rule for cosec^2 leads you to the point I was at anyway right?

Of course it does / it must by the pythagoren identity. The reverse chain is not cosec^3 though. You should be doing
cosec^2 <-> (cosec cot) cosec
so the right hand side (derivative) is a cubic, the integral on the left is a square which is becuse the derviative of 1/sin is cos/sin^2 (+c, and negated).

Why is it not cosec^3 because its what ive done for every question and its been okay.