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If your mother is still coughing then either the drug treatment was not effective or there is some other cause. A repeat chest X-ray should be done and if need be a CT scan.
A cough is generally a result of an irritant such as stomach acid (gastric reflux), mucus (asthma), hair spray, perfume, dust, pollen, mold, even spicy food, post nasal drip, or infection. It could also be due to infection such as pneumonia or bronchitis. Chronic sore throats can be allergic, viral or bacterial in origin. Many viral conditions like EBV (glandular fever) behave in this way with recurrent respiratory infections. Many parasitic infestations and worms increase the eosinophil counts in the blood and thus cause an increased risk of upper respiratory infections. Conditions like sarcoidosis, asbestosis and silicosis along with pulmonary fibrosis should be ruled out.
Since I cannot examine your mother and know other related conditions she may be having, nor is a detailed history possible on net, I have listed the various possibilities that should be looked into. Please consult your PCP to run tests and examination to clinch a diagnosis. Hope this helps. Take care!