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Diane, a long-standing cough in cats is usually due to feline asthma. A short-term cough can arise with feline herpsevirus (FHV-1) infection involving the trachea or pharynx (tracheitis or pharyngitis, respectively). Please have his chest X-rayed at your earliest convenience. A complete blood count can be helpful as well. I look for an elevation in the white blood cell eosinophil commonly seen in asthmatics.
Initial therapy for asthmatics usually involves a glucocorticoid (steroid) such as prednisolone. A bronchodilator might be of value in reducing the dose of prednisolone needed to keep my patient comfortable. Please respond with further questions or concerns if you wish.