After rotating 60 degrees counterclockwise around the right-angled vertex A,

∠EAC = 60 degrees

At the same time AE = AC.

Therefore, the triangle EAC is an isosceles triangle with the vertex angle ∠EAC = 60 degrees, so it is also an equilateral triangle.

EC = AC

ABC is an isosceles right triangle, so

AC = AB

therefore

EC = AB = 1