In this manuscript, we investigate and study some cohomological properties of Banach algebras. Let A be a Banach algebra with a bounded left approximate identity, and let B be a Banach A - bimodule. We show that if AB** and B ** A are subset of B, then H¹(A,B(2n+1)) = 0 for all n = 0, whenever H¹(A,B *) = 0.