how to simplifying this euler expression?

Question

given this equation: $(1-2e^{-jw} + e^{-2jw})$

how does that simplify to this? $(1-e^{-jw})^2$

I'm not sure what algebraic steps to take to get the 2nd equation.

Answer 1

It's just the binomial square formula: $(a-b)^2 = a^2+b^2-2ab$, where in this case, $a=1$ and $b=e^{-jw}$. $(1-e^{-jw})^2=1^2+(e^{-jw})^2-2e^{-jw}=1+e^{-2jw}-2e^{-jw}$.

Answer 2

Hint:

Substutute $u=e^{-jw}$ to obtain: $$1-2u+u^2 \tag{1}$$ Therefore, you must obtain $(2)$ from $(1)$: $$(1-u)^2 \tag{2}$$ Can you do this?