Challenge Level

On a sphere of radius $R$ we use a scale factor and the equivalent formula is $$\cos {a\over R} = \cos {b\over R} \cos {c\over R}.$$

All the familiar trigonometric identities in Euclidean Plane
Geometry have their counterparts in Spherical Geometry (otherwise
known as Elliptic Geometry) and also in Hyperbolic Geometry. These
are the geometries on flat surfaces (Euclidean Geometry), on
surfaces of positive curvature (Spherical Geometry) and on surfaces
of negative curvature (Hyperbolic Geometry). See the article
Curvature of Surfaces.

Most of the results in these two other geometries are much more similar in form to the trigonometric identities you meet in school than the result proved here which is equivalent to Pythagoras' Theorem.

The corresponding Pythagorean Theorem for right-angled triangles in Hyperbolic Geometry is: $$\cosh a = \cosh b \cosh c.$$