Prove that $\tau(\Bbb{C}P^n)\oplus1=\eta\otimes\ldots\otimes\eta$ (n+1 times), where $1$ denotes the trivial complex line bundle over $\Bbb{C}P^n$, $\tau(\Bbb{C}P^n)$ is the complex tangent bundle of $\Bbb{C}P^n$ and $\eta$ denotes the Hopf bundle $p:S^{2n+1}\rightarrow\Bbb{C}P^n$ with fiber $F=S^1$.
I think I have to show that there is a fiber bundle equivalence, so I have to show that there is a common fiber. The fiber of the LHS is $\Bbb{C}^n\oplus\Bbb{C}=\Bbb{C}^{n+1}$ and the fiber of the RHS is ${S^1}\otimes\ldots\otimes{S^1}$. I don't understand how they can be equal since ${S^1}\otimes\ldots\otimes{S^1}$ seems to be bounded whereas $\Bbb{C}^{n+1}$ is not bounded. Also I don't understand how to take the tensor product of $S^1$ as it is not a vector space under normal addition and multiplication. Can someone help me understand?