Proof of $|1-\bar zw|=|1-z\bar w|$.

Question

I want to prove $|1-\bar zw|=|1-z\bar w|$.

The crux being $\forall z,w\in \mathbb C: \bar zw=z\bar w$

Work done so far: I tried breaking down $z$ and $w$ into its components and brute-force my way to validate the equality, but I was not able to prove it.

Any help will be greatly appreciated.

Answer 1

Hint:

We have $|\bar{x}|=|x|$. You just have to identify what should be the $x$ in your question.

Also, in general, $\bar{z}w \ne z\bar{w}$.