= Solution
We induct on $n$. The claim is immediate for $\mathbb{CP}^0$. In the stated cover, $U=U_0\cong\mathbb C^n$ is contractible. The homotopy
$$
[z_0:z_1:\cdots:z_n]\longmapsto[t z_0:z_1:\cdots:z_n]
$$
deformation retracts $V$ onto the hyperplane $z_0=0$, which is $\mathbb{CP}^{n-1}$. Moreover,
$$
U\cap V\cong\mathbb C^n\setminus\{0\}
$$
deformation retracts onto $S^{2n-1}$.
The <Mayer-Vietoris theorem>[Mayer--Vietoris sequence] for the de Rham complex contains
$$
H^{i-1}_{\mathrm{dR}}(U\cap V)
\longrightarrow H^i_{\mathrm{dR}}(\mathbb{CP}^n)
\longrightarrow H^i_{\mathrm{dR}}(U)\oplus H^i_{\mathrm{dR}}(V).
$$
For odd $i>1$, the group on the right vanishes by contractibility and the induction hypothesis, while the group on the left vanishes because $i-1$ is positive and even and is neither $0$ nor $2n-1$. Thus the middle group vanishes. For $i=1$, the preceding map
$$
H^0(U)\oplus H^0(V)\longrightarrow H^0(U\cap V)
$$
is surjective because all three spaces are connected, so the connecting map into $H^1(\mathbb{CP}^n)$ is zero. Therefore all odd de Rham cohomology groups of $\mathbb{CP}^n$ vanish.
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