We have to prove the following identity:
$$z_1 \bar{z_2} = \frac{1}{4}(|z_1 + z_2|^2 + i|z_1 + iz_2|^2 - |z_1 - z_2|^2 - i|z_1 - iz_2|^2)$$
It says to use the identity we just proved, which is $\Re(z_1 \bar{z_2}) = \frac{1}{4}(|z_1 + z_2|^2 - |z_1 - z_2|^2)$
I get that we consider the identity we have proven with $\bar{z_2}$ and also with substituting $\bar z_2 \to i \bar z_2$. We also use the fact that $z = \Re(z) + i \Im(z)$, but I'm confused to the signs: Since $\Re(iz) = - \Im (z)$, shouldn't the terms multiplied by $i$ have the opposite sign than they do in the identity?
Note that
$$\Im\{z_1\bar{z_2}\}=\Re\{-iz_1\bar{z_2}\}=\Re\{z_1\overline{iz_2}\}=\frac14(|z_1+iz_2|^2-|z_1-iz_2|^2)$$
So the identity in your question is actually correct.