\(\displaystyle \Re{(z)} = a,\text{ so let }z = a +ib.\)
\(\displaystyle |z| =\sqrt{ a^{2}+b^{2}}=1, \\ \text{ so }a^{2}+b^{2}=1,\\ b^{2}=1-a^{2}.\)
\(\displaystyle z^{2}=(a+ib)(a+ib)=a^{2}-b^{2}+2iab,\\ \text{so }\Re{(z^{2})}=a^{2}-b^{2}=a^{2}-(1-a^{2})=2a^{2}-1.\)
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