I Have Have A Recurring Cough For About 4 Months

I have have a recurring cough for about 4 months now. Not a heavy cough just a light cough that there doesn't seem to be any reason for. I m not stuffed up, no pain, just this darn cough.
I went to a walk in clinic today and the nurse said chest sounded clear, blood pressure 120/70, no problems in ears or throat ? She prescribed a $60 cough syrup ? I want to cure the problem not just delay it.....
No flem or not much flem but that might be getting alittle worse these last couple of weeks.....
What should I do next ?