I am a bit rusty on my de Rham cohomology, and I'm hoping that someone here could help me.

I want to find the cohomology of $T^*\mathbb{CP}^n$ (seen as a real manifold). Now, this should be equal to the cohomology of $\mathbb{CP}^n$ since the two are homotopic (by homotopy of each fibre with a point), thus the problem reduces to the computation of $H^\bullet(\mathbb{CP}^n)$. How can I proceed to find it? Would something as the third possibility proposed in this answer work (by taking $G=\mathbb{C}^*$ acting on $\mathbb{C}^{n+1}$)?