In one course I had to solve the following exercise showing that cup product and connecting homomorphism "commute" in the way that statement makes sense:
Let $A\subseteq X$ be topological spaces. Consider the connecting homomorphism $\partial: H^{\ast}(A)\rightarrow H^{\ast +1}(X,A)$ and the inclusion $i: A\rightarrow X$. Then for $\alpha\in H^{\ast}(A)$, $\beta \in H^{\ast}(X)$: $$\partial (\alpha \cup i^{\ast}\beta)=\partial\alpha \cup \beta.$$
That is all nice and good, but what is the relevance of this result? What does it imply for the long exact sequence of cohomology?