Jacobian matrix of the parametrization of (part of) a ball

Question

I read (in E. Sernesi, Geometria 2) that the function $\varphi:\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times\mathbb{R}^{n}\to\mathbb{R}^{n+1}$ defined by $$\varphi(\theta_1,\ldots,\theta_n,r)=\left(r\prod_{k=1}^n\cos\theta_k,r\sin\theta_n\prod_{k=1}^{n-1}\cos\theta_k, r\sin\theta_{n-1}\prod_{k=1}^{n-2}\cos\theta_k,\ldots ,r\sin\theta_1\right)$$has an invertible Jacobian matrix. How can it be proved? I have calculated such matrix as $$\begin{pmatrix}\prod_k\cos\theta_k & -r\sin\theta_1\prod_{k\ne 1}\cos\theta_k &\ldots& -r\sin\theta_n\prod_{k\ne n}\cos\theta_k\\\sin\theta_n\prod_{k<n}\cos\theta_k& -r\sin\theta_1\sin\theta_n\prod_{k<n,k\ne 1}\cos\theta_k&\ldots&-\cos\theta_n\prod_{k<n,k\ne n}\cos\theta_k\\\vdots& \vdots &\ddots&\vdots \\ \sin\theta_1&r\cos\theta_1 &\ldots&0\end{pmatrix}$$I have tried to use induction to prove it, and showed that it holds for $n=2$, and $n=1$, by calculating the determinant, but I do not know how to prove it for the general case. Using the determinant in the general case by using the Laplace expansion does not seem applicable to me... verifying the independence of the columns or rows may be an option, but I cannot do so. I thank any answerer very much!