Isomorphic pairs of quadratic forms

Let k be a field of characteristic distinct from 2, V a finite dimensional vector space over k. We call two pairs of quadratic k-forms (f1,g1), (f2,g2) on V isomorphic if there exists an isomorphism s:V→V such that f2=f1∘s, g2=g1∘s. We prove that if f1+tg1≃f2+tg2 over k(t) and either the form f1+tg1 is anisotropic, or det(f1+tg1) is a squarefree polynomial of degree at least dimV−1, then the pairs (f1,g1) and (f2,g2) are isomorphic.