  We investigate the question of how to compute the cotensor product, and more
generally the derived cotensor (i.e., Cotor) groups, of a tensor product of
comodules. In particular, we determine the conditions under which there is a
K\"{u}nneth formula for Cotor. We show that there is a simple K\"{u}nneth
theorem for Cotor groups if and only if an appropriate coefficient comodule has
trivial coaction. This result is an application of a spectral sequence we
construct for computing Cotor of a tensor product of comodules. Finally, for
certain families of nontrivial comodules which are especially topologically
natural, we work out necessary and sufficient conditions for the existence of a
K\"{u}nneth formula for the $0$th Cotor group, i.e., the cotensor product. We
give topological applications in the form of consequences for the $E_2$-term of
the Adams spectral sequence of a smash product of spectra, and the Hurewicz
image of a smash product of spectra.
