Okay this has been driving me crazy and I cant find my error.

Okay here it is.

So first Since sine is periodic and it is squared

So if I parameterize the unit circle in in the complex plane with

Since

Now if I sub z in I get

Here I use the binomial theorem to expand the numerator

Now by the cauchy integral theorem the only non zero integral is when k=n and it gives

So finally we have

So I checked this formula numerically and it works except for the

I know that is can't be correct becuase the integrand is always non negative.

Thanks for looking