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Here a direct approach. Recall the power-series \begin{equation*} \arccos(z) = \frac\pi2 - \sum_{k\ge0}\binom{2k}{k}\frac{z^{2k+1}}{4^k(2k+1)}. \end{equation*} From this series it is clear that $\arccos(x^Ty)$ is conditionally negative definite (because it is of the form "const $-$ positive definite").