The problem is to find the cohomology ring of the unitary group. The idea is to look at the fibration $U(n-1) \to U(n) \to S^{2n-1}$ and use the Leray-Hirsch theorem. But I don't understand why this theorem can be used here. Can you help me?
PS. In this question (Calculate the cohomology group of $U(n)$ by spectral sequence.) the same problem is solved with Serre spectral sequence but I'm a bit afraid of spectral sequences yet so I prefer a solution using only Leray-Hirsch theorem.
