<s>[INST] How to decompose this matrix exponential? I would some help with the steps to decompose the below matrix exponential. 
$\exp\left[ \zeta \left ( \begin{matrix}
-\cos(x) & i \sin(x) \\ 
-i \sin(x) & \cos(x)
\end{matrix} \right ) \right] = \left ( \begin{matrix}
\cosh(\zeta)-\sinh(\zeta)\cos(x) & i \sinh(\zeta)\sin(x) \\ 
-i \sinh(\zeta)\sin(x) & \cosh(\zeta)+\sinh(\zeta)\cos(x)
\end{matrix} \right )$
I think the first matrix could be factored as Pauli matrices but I have not had any luck trying this. 
 [/INST] First, check your equality when $\sin(x)=0$.
Second, notice that $\zeta\begin{pmatrix}-\cos(x)& i\sin(x)\\-i\sin(x)& \cos(x)\end{pmatrix}=\zeta\begin{pmatrix}i& 0\\ 0& 1\end{pmatrix}\begin{pmatrix}-\cos(x)& \sin(x)\\\sin(x)& \cos(x)\end{pmatrix}\begin{pmatrix}-i& 0\\ 0& 1\end{pmatrix}$ 
and $\begin{pmatrix}i& 0\\ 0& 1\end{pmatrix}\begin{pmatrix}-i& 0\\ 0& 1\end{pmatrix}=\begin{pmatrix}1& 0\\ 0& 1\end{pmatrix}$.
Next, $\zeta\begin{pmatrix}-\cos(x)& \sin(x)\\\sin(x)& \cos(x)\end{pmatrix}=\begin{pmatrix}1+\cos(x)& 1-\cos(x)\\-\sin(x)& \sin(x)\end{pmatrix}\begin{pmatrix}-\zeta& 0\\ 0& \zeta\end{pmatrix}\begin{pmatrix}\dfrac{1}{2}& \dfrac{-1+\cos(x)}{2\sin(x)}\\\dfrac{1}{2}& \dfrac{1+\cos(x)}{2\sin(x)}\end{pmatrix}$.
Notice that $\begin{pmatrix}1+\cos(x)& 1-\cos(x)\\-\sin(x)& \sin(x)\end{pmatrix}\begin{pmatrix}\dfrac{1}{2}& \dfrac{-1+\cos(x)}{2\sin(x)}\\\dfrac{1}{2}& \dfrac{1+\cos(x)}{2\sin(x)}\end{pmatrix}=\begin{pmatrix}1& 0\\ 0& 1\end{pmatrix}$.
Thus, you need to compute $\exp(R\begin{pmatrix}-\zeta& 0\\ 0& \zeta\end{pmatrix}R^{-1})=R\begin{pmatrix} e^{-\zeta}& 0\\ 0 & e^{\zeta} \end{pmatrix}R^{-1}$, where $R=\begin{pmatrix}i& 0\\ 0& 1\end{pmatrix}\begin{pmatrix}1+\cos(x)& 1-\cos(x)\\-\sin(x)& \sin(x)\end{pmatrix}$ and $R^{-1}=\begin{pmatrix}\dfrac{1}{2}& \dfrac{-1+\cos(x)}{2\sin(x)}\\\dfrac{1}{2}& \dfrac{1+\cos(x)}{2\sin(x)}\end{pmatrix}\begin{pmatrix}-i& 0\\ 0& 1\end{pmatrix}$.
In order to finish you must use the formulas for the $\cosh(\zeta)$ and $\sinh(\zeta)$.
</s>