How we can show in the sphere or disc,there is no essential simple closed curves ?
In the mapping class groups By Benson farb , definition of essential closed curve is : a closed curve is called essential if it is not homotopic to a point, puncture, or a boundary component."
First, the sphere and disc are both simply connected.
Second, a path connected space $X$ is simply connected if and only if every continuous function $f : S^1 \to X$ is homotopic to a point; see here.
Putting these together, since a simple closed curve is the image of a continuous function $f : S^1 \to X$, it follows that every simple closed curve is inessential.