Look for the horizontal asymptote of
(sin x)/(1 + cos x).

This function does not have horizontal asymptote.

The horizontal asymptotes are the values to which the function approaches as x becomes larger and larger (+/- infinity).

Then you have to find the limits of the function for x -> + and - infinity.

If the limit does not exist then there is not horizontal asypmtotes.

Indeed the limit of that function when x goes to +/- infinity does not exist, because the denominator will continue fluctating between 1 and 2, and the numerator will be fluctuating between -1 and 1.

This function fluctuates periodically from negative infinity to positive infinity, it does not have a limit when x grows so it does not have horizontal asymptotes.