I am working with the standard $n$-dimensional sphere $S^n = \{\mathbf{x} = (x_1, x_2, ..., x_{n+1}) \in \mathbb{R}^n \text{ s.t. } \Vert \mathbf{x} \Vert = 1 \}$. Let $U = S^n\setminus \{(0, 0, ..., 1)\}$ (the set of all points minus a pole) and let $V = S^n\setminus \{(0, 0, ..., -1)\}$.
We can define the stereographic projection for $U$:
\begin{align} &\varphi(\mathbf{x}) : U \rightarrow \mathbb{R}^n \\ &\varphi(\mathbf{x}) = \frac{1}{1 - x^{n+1}}(x_1,...,x_n) \end{align}
Similarly the stereographic projection for $V$:
\begin{align} &\psi(\mathbf{x}) : V \rightarrow \mathbb{R}^n \\ &\psi(\mathbf{x}) = \frac{1}{1 + x^{n+1}}(x_1,...,x_n) \end{align}
I would like to show that the coordinate charts $(U, \varphi)$ and $(V, \psi)$ are compatible. Two charts are compatible if $U \cap V = \emptyset$ (clearly not the case) or if the compositions $\varphi\circ\psi^{-1}$ and $\psi\circ\varphi^{-1}$ are both $C^{\infty}$: clearly, I must show the latter.
I have been able to compute the inverse of the stereographic projections:
\begin{align} &\varphi^{-1}(\mathbf{y}) : \mathbb{R}^n \rightarrow U \\ &\varphi^{-1}(\mathbf{y}) = (\frac{2y_1}{1 + \Vert \mathbb{y}\Vert^2},...,\frac{2y_n}{1 + \Vert \mathbb{y}\Vert^2}, 1 - \frac{2}{1 + \Vert \mathbb{y} \Vert ^2}) \end{align}
\begin{align} &\psi^{-1}(\mathbf{y}) : \mathbb{R}^n \rightarrow V \\ &\psi^{-1}(\mathbf{y}) = (\frac{2y_1}{1 + \Vert \mathbb{y}\Vert^2},...,\frac{2y_n}{1 + \Vert \mathbb{y}\Vert^2}, \frac{2}{1 + \Vert \mathbb{y} \Vert ^2} - 1) \end{align}
I could compose them, but that seems tedious to me, instead, I would like to use the fact that the composition of two $C^{\infty}$ function is also $C^{\infty}$. I have been able to show that $\varphi, \psi$ are $C^{\infty}$ as I can use the geometric series to quickly find a Taylor series for each component in order to show that the partial derivatives with respect to $x_{n + 1}$, and the other partials are easy as either I am taking the derivative of a linear function, or a constant function.
However, showing that the inverse functions are $C^{\infty}$ is tripping me up. I have to show that they have continuous partial derivatives of all orders, and I don't quite know how to write a nice Taylor series for the ${2}/{1 + \Vert \mathbb{y}\Vert^2}$ bit, it will have Hessians and tensors and all that, since it's a multivariable function.
What else could I do to show that the inverse is $C^{\infty}$? Or, is there an alternative strategy?