Solution: Recall that \[ \cosh x = \frac{e^x + e^{-x}}{2}, \text{ and } \sinh x = \frac{e^x - e^{-x}}{2}. \] So, we have \begin{align*} (\cosh x - \sinh x)^n & = \left( \frac{e^x + e^{-x}}{2} - \frac{e^x - e^{-x}}{2}\right)^n \\ & = \left( \frac{2e^{-x}}{2} \right) ^n \\ & = e^{-nx}. \end{align*} The RHS will be \begin{align*} (\cosh x - \sinh x) & = \frac{e^{nx} + e^{-nx}}{2} - \frac{e^{nx} - e^{-nx}}{2} \\ & = \frac{2 e^{-nx}}{2} \\ & = e^{-nx} \\ & = \left( \cosh x - \sinh x \right) ^n. \end{align*}