when FullSimplify can't canonicalize down to a single form, e.g., IIn[296]:= FullSimplify[%, H > 0 && H \[Element] Integers] Out[296]= {1/8 Sinh[H Log[3 + 2 Sqrt[2]]]^2, 1/16 (-1 + Cosh[H Log[17 + 12 Sqrt[2]]]), 1/8 Sinh[2 H ArcSinh[1]]^2, ChebyshevU[-1 + H, 3]^2} you could just subtract one from the rest and hope FullSimplify finds, in this case, three zeroes. Alternatively, you can really punish FullSimplify for its laziness: In[298]:= FullSimplify[%.{x - y, y - z, z - w, w - x}] Out[298]= 0 Proof: In[77]:= Reduce[ForAll[{x, y, z, w}, {a, b, c, d}.{x - y, y - z, z - w, w - x} == 0], {a, b, c, d}] Out[77]= b == a && c == a && d == a Or In[78]:= Resolve[ForAll[{x, y, z, w}, {a, b, c, d}.{x - y, y - z, z - w, w - x} == 0]] Out[78]= a - b == 0 && b - c == 0 && a - d == 0 && c - d == 0 --rwg