This exercise is concerned with stereographic projection from the unit sphere in \(\mathbb{R}^{d+1}\) onto the equatorial plane \(\mathbb{R}^{d}\). Latitude on the sphere is measured by the polar angle \(\theta\), starting from zero at the north pole, through \(\theta=\pi / 2\) at the equator up to \(\theta=\pi\) at the south pole. Every point on the sphere is a pair \(z=(e \sin \theta, \cos \theta)\) where \(e\) is a unit equatorial vector. The stereographic image of \(z\) is the point

\[
\omega=e \cot (\theta / 2)=e \cos (\theta / 2) / \sin (\theta / 2)
\]

so that the southern hemisphere is projected into the unit ball, and the northern hemisphere to its complement in \(\mathbb{R}^{d}\). Deduce that the stereographic image of the uniform spherical distribution is