Euclidean Distance on a Sphere

Consider the diagram:

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Using the identity $\cos(\theta)=1-2\sin^2(\theta/2)$, the distance is $$ 2r\sin(\theta/2)=r\sqrt{2-2\cos(\theta)} $$

Just draw a picture of the intersection of the sphere with a plane containing the center and the two points.

The euclidean distance is the length of a hypoteneuse of a triangle whose other two sides have lengths $R\cdot \sin(\theta_{12})$ and $R\cdot (1-\cos(\theta_{12}))$, respectively. Then apply Pythagorean theorem.