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For a complex number $z$, whose modulus $|z|=1$ (but it's not entirely real $z\neq1$), how do I show that

$$\Re\left[\frac{1}{1-z}\right] = \frac{1}{2}$$

I have tried applying the formula $|z|^2 = z\bar{z}$ given that i have seen a lot of similar problems solved this way and the fact that $\Re[x] = \frac{x+\bar{x}}{2}$ for $x \in \mathbf{C}$.

I can't seem to rearrange it properly however. Thanks in advance!

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Your idea is fine, indeed using that $\Re(w)=\frac12 (w+\bar w)$, we obtain

$$\Re\left[\frac{1}{1-z}\right] =\frac12\left(\frac1{1-z}+\frac1{1-\bar z}\right)=\\=\frac12\left(\frac{(1-\bar z)+(1-z)}{(1-z)(1-\bar z)}\right)=\frac12\left(\frac{2-z-\bar z}{2-z-\bar z}\right)=\frac 12$$

As an alternative, by $z=e^{i\theta}$ and Euler's identity

$$\frac{1}{1-z}=\frac{1}{1-e^{i\theta}}\frac{1-e^{-i\theta}}{1-e^{-i\theta}}=\frac{1-e^{-i\theta}}{2-e^{i\theta}-e^{-i\theta}}=\frac{1-\cos \theta+i\sin \theta}{2(1-\cos \theta)}=\frac12+i\frac{\sin \theta}{2(1-\cos \theta)}$$

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