Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 6 / 4 / b / Solution

If either vector vanishes, both arguments are zero. Otherwise choose a rotation matrix $R$ sending $x/|x|$ to $y/|y|$. The surface measure on a sphere is rotation invariant. Changing variables $\sigma =R\omega$ givesThus the two spherical integrals are equal. The property extends to complex continuous $\varphi$ by treating real and imaginary parts separately, as needed for the Fourier exponential.