What is the ruling on studying and learning Pythagoras's theorem in mathematics, and is there any sin incurred for having studied it in the past?

There is no Shariah impediment to studying and learning the Pythagorean theorem in triangles. It is one of the permissible worldly sciences. It states that the sum of the squares of the lengths of the two sides adjacent to the right angle in a right-angled triangle equals the square of the length of the hypotenuse.