Working through a paper and cannot seem to confirm the following equality:
$\exp(-\beta\hbar\omega_i\hat{S_{iz}}) = \cosh(\frac{\beta\hbar\omega_i}{2})-2\hat{S_{iz}}\sinh(\frac{\beta\hbar\omega_i}{2})$
where
$\exp(-\beta\hbar\omega_i\hat{S_{iz}}) = \exp(-\beta\hbar(\omega_1\hat{S_{1z}}+\omega_2\hat{S_{2z}}))$,
$\hat{S_{1z}} = \begin{matrix} 1/2 & 0 & 0 & 0 \\ 0 & 1/2 & 0 & 0 \\ 0 & 0 & -1/2 & 0 \\ 0 & 0 & 0 & -1/2 \end{matrix}$
and
$\hat{S_{2z}} = \begin{matrix} 1/2 & 0 & 0 & 0 \\ 0 & -1/2 & 0 & 0 \\ 0 & 0 & 1/2 & 0 \\ 0 & 0 & 0 & -1/2 \end{matrix}$
Any help/resources would be appreciated. My main concern is how the matrix moves in front of the hyperbolic sin function and out of the argument.