Here's an oddity that you might not know about. Placebo pills, those sugar pills used in studies to trick subjects into thinking they are taking "real medicine," are working better and better over time. No one really knows why, but it likely has to do with the success of drug marketing. Drug commercials do what they are supposed to do: they train our brains to associate taking pills with feeling better in the future.
Once the whole population is trained into a Pavlovian response to pills that way, then not only do people respond to drugs better overall, but subjects in trials who get placebos enjoy a conditioned improvement in symptoms as well.
This phenomenon is a multi-billion dollar frustration for the drug industry. In order to get approval by the FDA, it has to show its drugs significantly outperform placebos when studied in clinical trials. With placebos working better all the time, the drug companies have to get more and more creative, twisting and even outright suppressing data to make drugs look better than duds.
It's worth keeping this in mind if you ever fill a prescription. Your belief that the drug will help your symptoms might be doing as much or more than the drug itself.
And in the meantime, take advantage of the placebo effect when your kid has a cough that won't go away. But as a matter of principle, I recommend you use a fish oil chewable rather than a sugar pill as your placebo.