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No, hypothyroidism can not itself cause a persistent cough or hoarseness, sometimes if the thyroid gland is physically enlarged it can cause shortness of breath, difficulty swallowing and a cough. But hoarseness is only caused when the recurrent laryngeal branch is involved in severe cases. If you do not have an enlarged/visible thyroid gland then hypothyroidismt will not cause hoarseness or a cough.
However, GERD can cause both hoarseness and a persistent cough due to acid reflux and irritation of the throat from inside. It can also cause repsiratory tract infections. Pulmonary hypertension can cause a cough but not hoarseness.
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