Multiply uv together, expand, and you get uv = -1. Since you now know u+v=-1 and uv=-1, u and v are roots of the quadratic x^2+x-1=0. Use the quadratic formula and you will find that |u-v|=sqrt5. Then using your trig expression and the fact that cos2pi/5>cos4pi/5, you know that u-v>0, therefore u-v=sqrt5 as required.
