In the notebook below, I calculate a general expression and get zero. This is wrong in general because of the following special case: for qt = 2q, I get the sine of some phases. I have confirmed the special case with pen and paper. Can somebody please tell me why Mathematica screws up like this?

P.S. I know that the attached notebook was created in Mathematica 11. I assure you that Mathematica 13 makes the exact same mistake.