Summary: We prove a local version of a recently established theorem by Myroshnychenko, Ryabogin and the second named author [S. Myroshnychenko et al., Int. Math. Res. Not. 2019, No. 10, 3015–3031 (2019; Zbl 1419.52005)]. More specifically, we show that if \(n\geq 3\), \(g:\mathbb{S}^{n-1}\to \mathbb{R}\) is a bounded measurable function, \( U\) is an open connected subset of \(\mathbb{S}^{n-1}\) and the restriction (section) of \(f\) onto any great sphere perpendicular to \(U\) is isotropic, then \(\mathcal{C}(g)\vert_U=c+\langle a,\cdot \rangle\) and \(\mathcal{R}(g)\vert_U=c'\), for some fixed constants \(c,c'\in \mathbb{R}\) and for some fixed vector \(a\in \mathbb{R}^n\). Here, \( \mathcal{C}(g)\) denotes the cosine transform and \(\mathcal{R}(g)\) denotes the Funk transform of \(g\). However, we show that an even \(g\) does not need to be equal to a constant almost everywhere in \(U^\perp \operatorname{:=} \bigcup_{u\in U}(\mathbb{S}^{n-1}\cap u^\perp )\). 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.