Inferior thyroid artery ligation during thyroidectomy is done
For recurrent Laryngeal nerve
Superior laryngeal nerve
Vagus nerve
Glossopharyngeal nerve
Inferior thyroid artery:
It is a branch of thyrocervical trunk from the first part of the subclavian artery. It first runs upwards along the medial border of  scalenus anterior, and then passes medially behind the carotid sheath to reach the back of the thyroid lobe, where it is intimately related to the recurrent laryngeal nerve. The recurrent laryngeal nerve presents a variable relationship with the artery. It may pass behind or in front of the loop of the artery or between the branches of the artery. The artery gives 4 or 5 branches.
KEY CONCEPT:
During thyroidectomy, the superior thyroid artery is ligated near the gland to save the external laryngeal nerve; and the inferior thyroid artery is ligated away from the gland to save the recurrent laryngeal nerve.