I'm beeing stuck with proving
$$T\mathbb{C}P^n\oplus(\mathbb{C}\times\mathbb{C}P^n)\cong\oplus_{i=1}^{n+1}\mathcal{O}(1).$$
The keypoint seems to be the identity
$$T\mathbb{C}P^n\cong\text{Hom}(\mathcal{O}(-1),\mathcal{O}(-1)^\perp),$$ where $\mathcal{O}(-1)^\perp$ denotes an orthogonal complement of $\mathcal{O}(-1)$ in the trivial bundle of rank $n+1$.
In the real case there is analogously
$$T\mathbb{R}P^n\oplus(\mathbb{R}\times\mathbb{R}P^n)\cong\oplus_{i=0}^{n}\gamma^1(\mathbb{R}^n)$$
With again the keypoint being the (more general) identity. $$TGr_n(\mathbb{R}^{n+k})\cong\text{Hom}(\gamma^n(\mathbb{R}^{n+k}),\gamma^n(\mathbb{R}^{n+k})^\perp).$$
Given the last identity the real case gets quite simple as
$$T\mathbb{R}P^n\oplus(\mathbb{R}\times\mathbb{R}P^n)\cong\text{Hom}(\gamma^1(\mathbb{R}^{n+1}),\gamma^1(\mathbb{R}^{n+1})^\perp)\oplus\text{Hom}(\gamma^1(\mathbb{R}^{n+1}),\gamma^1(\mathbb{R}^{n+1}))\cong\text{Hom}(\gamma^1(\mathbb{R}^{n+1}),\mathbb{R}P^n\times\mathbb{R}^{n+1})\cong\oplus_{i=0}^{n}\gamma^1(\mathbb{R}^n)$$ making extensive use of the fact, that every real fiberbundle is isomorphic to it's dual.
In the complex case I don't know how to adapt this strategy as we don't have the selfduality as in the real case. Also I don't know how to prove the two key identities. Thank you in advance for your help.
$\newcommand{\C}{\Bbb C}$ This is not true in algebraic geometry, but if you are working in differential geometry then every short exact sequence of vector bundles split after choosing an hermitian metric. So we just need to find a sequence like $$ 0 \to \mathcal O_X \to \mathcal O_X(1)^{n+1} \to TX \to 0$$
But in fact this sequence is exact and well known, called the "Euler sequence". The first map is sending $n+1$ scalars $a_0, \dots, a_n$ to the vector fields $X = \sum a_i \frac{\partial}{\partial x_i}$. This vector fields is precisely zero when it has radial direction, that is where $X$ is the multiple of the Euler vector fields $E = \sum \frac{\partial }{\partial x_i}$.
The name comes from the Euler relation $Ef = \deg(f) f$ if $f$ is an homogenous polynomial.
You should be able to find more informations about this sequence in the book of Hartshorne, algebraic geometry; or the book by Milnor and Stasheff on characteristic classes.