I had a lecture last week which dealt with the Frobenius Endomorphism on elliptic curves. The lecturer showed an example at the end of the lecture, when almost out of time and I don't quite understand it. We had the elliptic curve $E: Y^2+XY=X^3+X$ over $\mathbb{F}_2$. He then wrote the following: $[2](X,Y)=(X,Y)+(X,Y)=(X^2+\frac{1}{X^2},1+\frac{1}{X^2}+Y^2+\frac{Y^2}{X^4})=\text{Frob}(X+\frac{1}{X},1+\frac{1}{X}+Y+\frac{Y}{X^2})$
Now I can understand the last equal sign, since it seems the Frobenius endomorphism just raises everything to the (in this case) second power. However, I don't really know how we deduce the other parts. It might be that the teacher skipped quite a few steps, so I don't quite get it. Anybody any tips (wikipedia isn't much help).