If you are coughing after a deep breath on room air, it is usually indicative of inflamation in the lungs, I do the same thing and take prednisone it usually goes away.
Hello and hope you are doing well.
Cough basically is a mechanism to clear the secretions from the lungs to prevent collection and infection. If its cold air which is inhaled, it can sometimes irritate the throat and cause cough on inspiration.
Hope this helped and do keep us posted.