In general, air conditioning does not cause coughs. AC just cools incoming air from the outside and brings it into the house. Typically this air is brought through a filter which catches dust and allergens making the AC air even more clean. However, if this filter is missing, or has become dysfunctional, then breathing in this air can irritate the lungs if you are susceptible to whatever material is there.
I would say if your cough is related to the AC that is not properly filtered, then it is either infectious or allergic in nature. If you are being exposed to a virus, then you may develop a lung infection called bronchitis
. This type of infection lasts for a few weeks and can cause a very nagging cough. If the infection is allergic, then you may be breathing in material that is causing irritation to your lungs resulting in a cough. These two can be difficult to distinguish without seeing a doctor
The best type of physician for you to see about this problem is a primary care physician
. He or she can take a more detailed history and perform a thorough physical exam. You can discuss the most likely causes of your cough in the context of your medical history. I hope you get to the bottom of it soon.