Hello Welcome in H.C.M. I have gone through your query regarding dry cough, ear pain and popping sound in ear after taking deep breath. Most probably your dry cough is due to throat infection (sore throat) and this throat infection is obstructing the eustachian tube which ultimately results blockade in ear and you feel it as popping sound during deep breathing.this is the conducted sound of breathing due to negative pressure in middle ear. To manage it you may take- 1- Steam inhalation. 2- Anti histaminic. 3- Decongestant. 4- Tab Paracetamol. Probably this treatment will help you but any how if still your problem persist than consultation with ENT specialist will become necessary for clinical examination. Regards.