Given a coalgebra coextension, we introduce a filtration whose associated spectral sequence computes the Hochschild cohomology of the coextension. We then apply this cohomological machinery to compute the Hopf-cyclic cohomology of a quantized enveloping algebra of a Lie algebra. Time permitting, we will discuss as an application of this method, the Hopf-cyclic cohomology of Connes-Moscovici Hopf algebras. (Based on a joint work with A. Kaygun.)