Coordinate systems/Derivation of formulas/Cartesian from spherical unit vectors

From Wolfram Mathworld, we have the following relations for the unit vectors in a Spherical coordinate system:

$\begin{array}{cc}\widehat{𝐫}& =\sin \theta \cos \varphi \widehat{𝐱}+\sin \theta \sin \varphi \widehat{𝐲}+\cos \theta \widehat{𝐳}\\ \hat{\mathit{\theta }}& =\cos \theta \cos \varphi \widehat{𝐱}+\cos \theta \sin \varphi \widehat{𝐲}-\sin \theta \widehat{𝐳}\\ \hat{\mathit{\varphi }}& =-\sin \varphi \widehat{𝐱}+\cos \varphi \widehat{𝐲}\end{array}$

We first write this as a matrix equation:

$\left(\begin{array}{c}\widehat{𝐫}\\ \hat{\mathit{\theta }}\\ \hat{\mathit{\varphi }}\end{array}\right)=\left(\begin{array}{ccc}\sin \theta \cos \varphi & \sin \theta \sin \varphi & \cos \theta \\ \cos \theta \cos \varphi & \cos \theta \sin \varphi & -\sin \theta \\ -\sin \varphi & \cos \varphi & 0\end{array}\right)\left(\begin{array}{c}\widehat{𝐱}\\ \widehat{𝐲}\\ \widehat{𝐳}\end{array}\right)$

We can then find the inverse of the matrix to make $\widehat{𝐱},\widehat{𝐲},\widehat{𝐳}$ the subjects.

$\left(\begin{array}{c}\widehat{𝐱}\\ \widehat{𝐲}\\ \widehat{𝐳}\end{array}\right)=\left(\begin{array}{ccc}\sin \theta \cos \varphi & \cos \theta \cos \varphi & -\sin \varphi \\ \sin \theta \sin \varphi & \cos \theta \sin \varphi & \cos \varphi \\ \cos \theta & -\sin \theta & 0\end{array}\right)\left(\begin{array}{c}\widehat{𝐫}\\ \hat{\mathit{\theta }}\\ \hat{\mathit{\varphi }}\end{array}\right)$

Which was what we set out to show.