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
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