NIP for some pair-like theories

Generalising work of Berenstein, Dolich and Onshuus (Preprint 145 on MODNET Preprint server, 2008) and Günayd?n and Hieronymi (Preprint 146 on MODNET Preprint server, 2010), we give sufficient conditions for a theory T _P to inherit N I P from T, where T _P is an expansion of the theory T by a unary predicate P. We apply our result to theories, studied by Belegradek and Zilber (J. Lond. Math. Soc. 78:563?C579, 2008), of the real field with a subgroup of the unit circle.