= Solution
Hold $r$ fixed and differentiate the position <vector> $\mathbf X(\theta,\phi)$. Its coordinate tangent <vectors> are
$$
\mathbf X_\theta=r(\cos\theta\cos\phi,\cos\theta\sin\phi,-\sin\theta),\qquad
\mathbf X_\phi=r(-\sin\theta\sin\phi,\sin\theta\cos\phi,0).
$$
Their <dot product> is zero, and their squared <Euclidean norms> are $r^2$ and $r^2\sin^2\theta$. To first order, the displacement is $\delta\mathbf X=\mathbf X_\theta\delta\theta+\mathbf X_\phi\delta\phi+O((|\delta\theta|+|\delta\phi|)^2)$. Squaring gives
$$
\boxed{\delta s^2\simeq r^2(\delta\theta^2+\sin^2\theta\,\delta\phi^2)}.
$$
This is the <line element on a sphere>. The chord distance and the short surface <arc length> agree to the order retained here.
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