Proof using complex numbers Prove that $\left|\dfrac{z-w}{1-\bar{z}w}\right| = 1$ where $\bar{z}$ is conjugate of $z$ and $\bar{z}w\ne 1$ if either $|z| = 1$ or $|w| = 1$.
I used $|c_1/c_2| = |c_1|/|c_2|$ and multiply out with $z = x + iy$ and $ = a+ib$ but I am getting stuck near finish.

Use the polar representation of the numbers $z$ and $w$, to let $|z|$ and $|w|$ appear.
If you expand $|z-w|^2=|1-\overline zw|^2$, you will see desired simplifications.
