We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author. More specifically, we show that if n ≥ 3, g : S n−1 → R is an even bounded measurable function, U is an open subset of S n−1 and the restriction (section) of f onto any great sphere perpendicular to U is isotropic, then C(g)| U = c + a, • and R(g)| U = c ′ , for some fixed constants c, c ′ ∈ R and for some fixed vector a ∈ R n . Here, C(g) denotes the cosine transform and R(g) denotes the Funk transform of g. However, we show that g does not need to be equal to a constant almost everywhere in U ⊥ := u∈U (S n−1 ∩u ⊥ ). For the needs of our proofs, we obtain a new generalization of a result from classical differential geometry, in the setting of convex hypersurfaces, that we believe is of independent interest.