Cohomology rings of $S^3\times\mathbb{C} P^{\infty}$ and $S^1 \times \mathbb{C} P^{\infty}/(S^1 \times \{x_0\})$

Question

From Hatcher :

Show that these two spaces have isomorphic cohomology rings

Elsewhere on this website I have seen someone claim that the latter space is a reduced suspension and hence have trivial cup product. Whereas the first space has some ring coming from Kunneth formula which "doesn't seem" to have trivial multiplication, at least to me.

Can someone please elaborate.