Thank you for the additional information.
It is good that both the chest x-ray and examination were normal, as that would exclude the more serious concerns, such as certain chronic infections.
The most likely cause of a persistent minor cough is an allergy, which can cause inflammation of the upper or lower airways. Since the cough is exacerbated when breathing in, that would suggest some inflammation of the airways of the chest, a non-infectious form of bronchitis
It is also possible to develop a chronic bronchitis due to irritation of the airways, most often from cigarette smoke. However, people with chronic bronchitis from cigarettes would usually have some degree of shortness of breath with exertion, and the problem will usually have been a problem for longer than a couple months. The clinical definition of chronic bronchitis is that it causes symptoms for three months out of the year for two consecutive years.
At this point, it would be reasonable to first try an over the counter antihistamine
, such as Claritin or Zyrtec, to see if it eases the symptom. If this is ineffective, then the next consideration would be a steroid inhaler
, such as Flovent.
If you have any further questions, please let me know.