Answer
See explanations.
Work Step by Step
Step 1. Let $c=x+yi$, we have $c^2=x^2-y^2+(2xy)i$ and $\overline{c^2}=x^2-y^2-(2xy)i$
Step 2. We have $\overline{c}=x-yi$ and $(\overline{c})^2=x^2-y^2-(2xy)i$
Step 3. Thus we have $\overline{c^2}=(\overline{c})^2$