One of the main applications of the group structure on an elliptic curve is in number theory. If two points P, Q on an elliptic curve have rational coordinates then so does P*Q. In particular, you can take Q=P, so that the line PQ is the tangent at P. Then you only have to know one rational point P on the curve in order to generate others. This is a powerful method for finding families of solutions to cubic diophantine equations.
I wondered how that idea would play out on the circle. A rational point on the unit circle is given by a primitive pythagorean triple (p,q,r). The tangent at P has gradient –p/q, and the line through (1,0) with that gradient meets the circle at the point (x,y) given by , , telling you that is a primitive pythagorean triple.
Nothing new there, of course, but it's a neat illustration of how the method works.