### Instruction: Let $z$ be a nonreal complex number such that $|z| = 1.$  Find the real part of $\frac{1}{1 - z}.$.### Response: I know that $|z| = 1$ means that $z$ lies on the unit circle in the complex plane, so it has the form $z = \cos\theta + i\sin\theta$ for some angle $\theta.$
I want to find the real part of $\frac{1}{1 - z},$ so I need to write this fraction in a form that separates the real and imaginary parts.
To do that, I can multiply the numerator and denominator by the complex conjugate of $1 - z,$ which is $1 - \overline{z}.$
This will make the denominator a real number, since $(1 - z)(1 - \overline{z}) = 1 - z - \overline{z} + |z|^2 = 1 - 2\text{Re}(z) + 1 = 2 - 2\text{Re}(z).$
So, I have $\frac{1}{1 - z} = \frac{1 - \overline{z}}{(1 - z)(1 - \overline{z})} = \frac{1 - \overline{z}}{2 - 2\text{Re}(z)}.$
Now, I can substitute $z = \cos\theta + i\sin\theta$ and $\overline{z} = \cos\theta - i\sin\theta$ to get $\frac{1}{1 - z} = \frac{1 - \cos\theta + i\sin\theta}{2 - 2\cos\theta}.$
The real part of this fraction is the numerator without the imaginary term, divided by the denominator.
So, the real part is $\frac{1 - \cos\theta}{2 - 2\cos\theta}.$