  Let X be a smooth complete toric variety. We describe the
Altmann-Ilten-Vollmert equivariant deformations of toric varieties in the
language of Cox rings. More precisely we construct one parameters families of
deformations of X, such that the total space of the deformation is a T-variety
of complexity one defined by a trinomial equation, and the deformation map is
equivariant with respect to the torus action. Moreover we show that the images
of all these families via the Kodaira-Spencer map form a basis of the vector
space H1(X,T_X).
