Abstract
Let $A$ and $B$ be unital Banach algebras, $X$ be an unital $A-B-$module and $T$ be the triangular Banach algebra associated to $A, B$ and $X$. The structure of some derivations on triangular Banach algebras was studied by some authors. Note that despite the apparent similarity between derivations and biderivations and also inner derivations and inner biderivations, there are fundamental differences between them. Although there are some studying of biderivations on triangular Banach algebras, any of them do not completely determine the structure of biderivations on triangular Banach algebras. In this paper, we completely characterize biderivations and inner biderivations from $T\times T$ to $T^*$ and we show that the first bicohomology group $BH^1(T, T^*)$ is equal to $BH^1(A, A^*)\oplus BH^1(B, B^*)$.