Let \(A=\ell^1(B)\) be the semigroup algebra of \(B\), the bicyclic semigroup. In [\textit{S. Bowling} and \textit{J. Duncan}, Semigroup Forum 56, No. 1, 130--145 (1998; Zbl 0910.46055)], the first simplicial cohomology group \(H^1(A,A^*)\) was shown to be isomorphic to \(\ell^\infty(\mathbb{N})\). In the paper under review, the authors give a resolution of \(\ell^\infty(B)\) which simplifies the computation of the cohomology of dual \(A\)-bimodules. This resolution is applied to show that the simplicial cohomology \(H^n(A,A^*)\) vanishes for \(n\geq 2\). It is also shown that the cyclic cohomology groups \(HC^n(A,A^*)\) vanish when \(n\) is odd and are one-dimensional when \(n\) is even (\(n\geq 2\)).