I do not know the answer. But possibly having the implicit form might help for looking it up.

  tpolys = TrigExpand[{Sin[u]*Sin[v]*Sin[v + u], Cos[u]*Cos[v]*Cos[v + u], Cos[u]*Sin[v]*Sin[v + u]}];
    tp2 = tpolys /. {Cos[u] -> cu, Cos[v] -> cv, Sin[u] -> su, Sin[v] -> sv};
    gb = GroebnerBasis[
       Join[{x, y, z} - tp2, {cu^2 + su^2 - 1, cv^2 + sv^2 - 1}], {x, y, z}, {cu, cv, su, sv}];
    implicit = First[gb]

    (* Out[154]= x^2 y + x^2 z - z^2 + y z^2 + z^3 *)

Thank you Vitaliy. Unfortunately I'm using Mathematica 9 and it doesn't have this feature. I will do some more digging around and post here if I find it.