Let \(\displaystyle{x\,,y\in S^{m1}}\) such that \(\displaystyle{d(x,y)=\theta}\).
Consider the perpendicular to \(\displaystyle{xy}\)  line from the origin. Then,
\(\displaystyle{\sin\,\dfrac{\theta}{2}=\dfrac{xy_{2}}{2}\iff xy_{2}=2\,\sin\,\dfrac{\theta}{2}}\).
Yes, the function \(\displaystyle{d}\) is a metric in \(\displaystyle{S^{m1}}\) and with this metric,
the triplet \(\displaystyle{\left(S^{m1}\,,d\,,\sigma\right)}\), where,
\(\displaystyle{\sigma(A)=\dfrac{\left\{s\,x\in \mathbb{R}^m\,,0\leq s\leq 1\right\}}{B_{2}^m}\,,\forall\,A\in\mathbb{B}(S^{m1})}\),
becomes a metric probability space.
