Problem
1) Find the Möbius transformation which maps the points $0,i,-i$ to $0,1,\infty$ respectively.
2) Prove that the image of the circle centered at $0$, of radius $1$ is the line $\{Re(z)\}=1$.
In $1)$ I didn't have problems, the homographic transformation $T(z)$ which satisfies the conditions given is $T(z)=\dfrac{2z}{z+i}$.
I don't know how to solve $(2)$. If I denote the circle by $C$, I want to show that $T(C)=\{Re(z)=1\}$. I've tried to prove the two inclusions of these sets but I couldn't, I would appreciate some help.
An element of $C$ is $e^{it}$ with $t\in\mathbb{R}$: $$T(e^{it})=\frac{2e^{it}}{e^{it}+i}=\frac{2e^{it}(e^{-it}-i)}{2+2\sin(t)}=\frac{2-2ie^{it}}{2+2\sin(t)}=\frac{1-i(\cos(t)+i\sin(t))}{1+\sin(t)}=$$ $$=\frac{1+\sin(t)-i\cos(t)}{1+\sin(t)}=1-i\frac{\cos(t)}{1+\sin(t)}\ .$$ This is a parametrization of the line $Re(z)=1$.