  We entirely compute the cohomology for a natural and large class of
$\mathfrak{osp}(1|2)$ modules $M$. We study the restriction to the
$\mathfrak{sl}(2)$ cohomology of $M$ and apply our results to the module
$M={\mathfrak D}_{\lambda,\mu}$ of differential operators on the super circle,
acting on densities.
