Why does my left lung hurt when i cough?

Cough: This is likely due to a strained muscle due to the coughing itself. Some can even have a small fracture of a rub from coughing so much/hard. If you feel short of breath, then you should be seen, otherwise this should get better as you start coughing less!

Left lung hurts: the lung itself won't hurt, the lung covering, pleura, would, a condition called 'pleurisy', and pain would be sharp stabbing in nature with breathing, there are several causes for pleurisy, the most common are viral infections of the pleura itself or, pneumonia, but then you would have other symptoms too. You may very well have chest wall pain, see ur doctor if no improvement with pain killers

Well : In general this kind a symptom is called pleurisy and can be a sign of a blood clot in your lung, infection or viral infection or nonspecific inflammation of the plura which is the lining or sack around a lung.Seek immediate medical attention