$$ \begin{split} \sqrt2 - \frac1{\sqrt2 - \dfrac1{\sqrt2 - \dfrac1{\sqrt2 - 1}}} &= \sqrt2 - \frac1{\sqrt2 - \dfrac1{\sqrt2 - \dfrac{\sqrt2 + 1}{2-1}}}\\ &= \sqrt2 - \frac1{\sqrt2 - \dfrac1{\sqrt2 - \left(\sqrt2 + 1\right)}}\\ &= \sqrt2 - \frac1{\sqrt2 + 1}\\ &= \sqrt2 - \left(\sqrt2 - 1\right)\\ &= 1 \end{split} $$
Hi, I do not understand the first step we have come to. I tried to lead to a common denominator but ended up confused. Please show how to come to the first step of solving this example.
The idea is to remember that $a^2-b^2=(a+b)(a-b)$ so you get $$ \frac{1}{\sqrt2-1} = \frac{1}{\sqrt2-1} \times \frac{\sqrt2 + 1}{\sqrt2+1} = \frac{\sqrt2 + 1}{\left(\sqrt2\right)^2 - 1^2} = \frac{\sqrt2 + 1}{2 - 1} = \sqrt2 + 1 $$