  We present solutions to additive and multiplicative Cousin problems
formulated on an axially symmetric domain $\Omega \subset \mathbb H$ for
slice--regular functions starting from the solutions for subclasses, namely
slice--regular slice--preserving functions and functions in a given vectorial
class. Consequently, we prove the vanishing of the corresponding cohomology
groups with respect to axially symmetric open coverings (Theorems 1.1, 4.1,
4.2). The primary tool used in the proofs of these theorems is the existence of
quaternionic Cartan coverings and Cartan's splitting lemmas. As an application,
we prove a jet interpolation theorem (Theorem 1.2) and show that every divisor
is principal (Theorem 5.1).
