Consider the relationships between Cartesian and spherical coordinates as explained in this webpage.
Cartesian coordinates are given in terms of spherical coordinates according to the following relationships:
$$ \begin{array}{rcl} x(r,\theta,\phi)&=&r\cos\theta\sin\phi\\ y(r,\theta,\phi)&=&r\sin\theta\sin\phi\\ z(r,\theta,\phi)&=&r\cos\phi \end{array} $$
Spherical coordinates are given in terms of Cartesian coordinates according to the following relationships:
$$ \begin{array}{rcl} r(x,y,z)&=&\sqrt{x^2+y^2+z^2}\\ \theta(x,y,z)&=&\arctan\left(\frac{y}{x}\right)\\ \phi(x,y,z)&=&\arccos\left(\frac{z}{\sqrt{x^2+y^2+z^2}}\right) \end{array} $$
Let us call $J$ the Jacobian matrix associated with the transformation from Cartesian to spherical coordinates as: $$ J = \frac{\partial(x,y,z)}{\partial(r,\theta,\phi)} = \begin{bmatrix} \cos\theta\sin\phi & -r\sin\theta\sin\phi & r\cos\theta\cos\phi \\ \sin\theta\sin\phi & r\cos\theta\sin\phi & r\sin\theta\cos\phi \\ cos\phi & 0 & -r\sin\phi \end{bmatrix} $$
Let us call $J'$ the Jacobian matrix associated with the transformation from spherical to Cartesian coordinates as $$ J' = \frac{\partial(r,\theta,\phi)}{\partial(x,y,z)} = \begin{bmatrix} \frac xr & \frac yr & \frac zr\\ \frac{-y}{r^2-z^2} & \frac{x}{r^2-z^2} & 0 \\ \frac{xz}{r^2\sqrt{r^2-z^2}} & \frac{yz}{r^2\sqrt{r^2-z^2}} & \frac{-x^2-y^2}{r\sqrt{r^2-z^2}} \end{bmatrix} $$
After transforming $J'$ into spherical coordinates by replacing the set $(x,y,z)$ variables, it becomes
$$ J' = \begin{bmatrix} \cos\theta\sin\phi & \sin\theta\sin\phi & \cos\phi\\ \frac{-\sin\theta}{r\sin\phi} & \frac{\cos\theta}{r\sin\phi} & 0 \\ \frac{\cos\theta\cos\phi}{r} & \frac{\sin\theta\cos\phi}{r} & \frac{-\sin\phi}{r} \end{bmatrix} $$
In a book on tensor calculus (Introduction to tensor analysis and the calculus of moving surfaces, P. Grinfeld), it is stated that the product $JJ'$ should amount to the identity matrix for arbitrary transformations in the Euclidean space. However, the product
$$ JJ'=\begin{bmatrix} \cos^2\theta\sin^2\phi & -r\sin^2\theta\sin^2\phi & r^2\cos\theta\cos^2\phi\\ \frac{-sin^2\theta}{r} & \cos^2\theta & 0\\ \frac{\cos\theta\cos^2\phi}{r} & 0 & \sin^2\phi \end{bmatrix} \neq \mathbf{I} $$
As we can see the above product is not the identity matrix. Therefore, what am I doing wrong?