My attempt at a solution:
Suppose that there exists a positive $\delta$ such that for each $x$ and each $y$ in $\mathbb R$, $|x-y|<\delta \implies |\sin e^x - \sin e^y|<1$. (Is $1$ a correct choice for $\epsilon$ here?)
We then try to derive a contradiction to show that $\sin e^x$ is not uniformly continuous on $\mathbb R$. Let $x_n=\ln (2n+\frac12)\pi$ and $y_n=\ln (2n+\frac32)\pi \space \space \space \space\space (n \in \mathbb N) $.
Is my attempt headed in the right direction? If so, I would appreciate any help sketching out the rest of the proof. If not, please correct my solution (for example, my choice of $x_n$ and $y_n$ seem wrong since they do not converge to any point in the domain) and perhaps provide some guidance.