In this paper, we study the nonconvex nonsmooth optimization problem (P) of minimizing a tangentially convex function with inequality constraints where the constraint functions are tangentially convex. This is done by using the cone of tangential subdifferentials together with a new constraint qualification. Indeed, we present a new constraint qualification to guarantee that Karush-Kuhn-Tucker conditions are necessary and sufficient for optimality of the problem (P). Moreover, various nonsmooth (generalized) constraint qualifications that are a modification of the well known constraint qualifications are investigated. Several illustrative examples are presented to clarify the connection between nonsmooth constraint qualifications and new constraint qualification.