For $(x,y) \in D(0,1)$ fixed, we define the complex number $z = x + iy$ and we set for $t$ real:
$$\mathrm{N}(x,y,t) = \frac{1 - |z|^2}{|z - e^{it}|^2} = \frac{1 - (x^2 + y^2)}{(x - \cos t)^2 + (y - \sin t)^2}$$
In the rest of this part, the pair $(x,y)$ is fixed in $D(0,1)$.
Show that $t \mapsto \mathrm{N}(x,y,t)$ is defined and continuous on $[0, 2\pi]$.