\(\left(\dfrac{\sqrt{a}-1}{\sqrt{a}+1}+\dfrac{\sqrt{a}+1}{\sqrt{a}-1}\right)\left(1-\dfrac{2}{a+1}\right)^2\)

\(=\left(\dfrac{\left(\sqrt{a}-1\right)^2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}+\dfrac{\left(\sqrt{a}+1\right)^2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right)\left(1-\dfrac{4}{a+1}+\dfrac{4}{\left(a+1\right)^2}\right)\)

\(=\left(\dfrac{\left(\sqrt{a}-1\right)^2+\left(\sqrt{a}+1\right)^2}{a-1}\right).\left(\dfrac{\left(a+1\right)^2}{\left(a+1\right)^2}-\dfrac{4\left(a+1\right)}{\left(a+1\right)^2}+\dfrac{4}{\left(a+1\right)^2}\right)\)\(=\dfrac{a-2\sqrt{a}+1+a+2\sqrt{a}+1}{a-1}.\dfrac{\left(a+1\right)^2-4\left(a-1\right)+4}{\left(a+1\right)^2}\)

\(=\dfrac{2a+2}{a-1}.\dfrac{a^2+2a+1-4a-4+4}{\left(a+1\right)^2}\)

\(=\dfrac{2\left(a+1\right)\left(a^2-2a+1\right)}{\left(a-1\right)\left(a+1\right)^2}\)

\(=\dfrac{2\left(a+1\right) \left(a-1\right)^2}{\left(a-1\right)\left(a+1\right)^2}\)

\(=\dfrac{2\left(a-1\right)}{a+1}\)