You can think of it this way. Observe that \(\displaystyle e^te^{-t}=e^{t-t}=e^0=1\). Thus, we see that either \(\displaystyle e^te^{-t}=1\implies e^{-t}=\frac{1}{e^t}\) or \(\displaystyle e^{t}=\frac{1}{e^{-t}}\). In our case, we want the latter result.