You can physically take these medications together, but the antibiotic is likely unnecessary. Dry coughs are typically caused by viral infections or allergies, which antibiotics cannot treat. Using antibiotics without a confirmed bacterial infection is not recommended. The cough syrup, which acts as an antihistamine and cough suppressant, is suitable for soothing a dry cough. I advise using only the cough syrup for temporary relief and avoiding the antibiotic unless it was specifically prescribed by a doctor. If your cough persists for more than a week, or if you develop a fever or difficulty breathing, please consult a healthcare professional for a proper evaluation.