Please check my proof
The original problem ask to determine sinx is uniformly continuous on$(-\infty, \infty )$ and if it is, Prove it
I think this function is uniformly continuous
Ok let do it .
Let $\epsilon ,\delta >0 $
$$|x-y|<\delta \rightarrow |f(x)-f(y)<\epsilon $$
$$ \rightarrow |sinx-siny|<\epsilon $$
Use identity
$$ \rightarrow |2cos(\frac{(x+y)}{2})sin(\frac{(x-y)}{2})|$$
$$ \rightarrow cos(\frac{(x+y)}{2})sin(\frac{(x-y)}{2})<\frac{\epsilon }{2}|$$
Choose $\delta =\frac{\epsilon }{2}$
then
$$ |x-y|<\delta \rightarrow 2|cos\frac{(x+y)}{2}sin\frac{(x-y)}{2}|<2\frac{\epsilon }{2}|=\epsilon $$
Then the function is uniformly continuous on$(-\infty ,\infty )$
