Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 39A / d / Solution

Taking the squared modulus and using ${\cosh }^{2}u=1+{\sinh }^{2}u$ givesThe incident wavevector makes an angle $\psi$ with the downward vertical, so $\sin 2\psi =2km/(k^2+m^2)$. Substitution gives the required formula.