Answer
$$\frac{x^2}{4}+(y-1)^2=1$$
See the attached graph.
Work Step by Step
Given: $$x=2cos\theta$$
$$y=1+sin\theta$$
$$ \implies cos^2\theta=\frac{x^2}{4}$$
and $$sin^2\theta =(y-1)^2$$
$$cos^2\theta+sin^2\theta=\frac{x^2}{4}+(y-1)^2$$
Thus, $$\frac{x^2}{4}+(y-1)^2=1$$
See the attached graph.