Prove directly from the definition of uniform continuity that $\displaystyle \sin(x)$ is uniformly continuous on $\displaystyle (-\infty,\infty).$
(Hint: a unit circle may help to make an estimation.)
Here are some tools that will help you.
$\displaystyle |\sin x-\sin y|=\left|2\sin\left(\frac{x-y}{2}\right)\cos\left(\frac{x+y}{2}\right)\right|$
$\displaystyle |\cos\theta|\leq1$ and you can use the unit circle to show that $\displaystyle |\sin\theta|\leq|\theta|$.
Let $\displaystyle \delta=\epsilon$ and don't look back.
The only thing is that this requires him to assume $\displaystyle \sin x$ is differentiable and furthermore that it's derivative is $\displaystyle \cos x$. (If they haven't proved that $\displaystyle \sin x$ is continuous yet, it's unlikely they've proved its derivative is $\displaystyle \cos x$.)