\(P=\left(\dfrac{\sqrt{a}}{2}-\dfrac{1}{2\sqrt{a}}\right)^2\cdot\left(\dfrac{\sqrt{a}-1}{\sqrt{a}+1}-\dfrac{\sqrt{a}+1}{\sqrt{a}-1}\right)\left(a>0;a\ne1\right)\)
\(=\left(\dfrac{\sqrt{a}\cdot\sqrt{a}}{2\sqrt{a}}-\dfrac{1}{2\sqrt{a}}\right)^2\cdot\left[\dfrac{\left(\sqrt{a}-1\right)^2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}-\dfrac{\left(\sqrt{a}+1\right)^2}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right]\)
\(=\left(\dfrac{a-1}{2\sqrt{a}}\right)^2\cdot\left[\dfrac{a-2\sqrt{a}+1-a-2\sqrt{a}-1}{\left(\sqrt{a}-1\right)\left(\sqrt{a}+1\right)}\right]\)
\(=\dfrac{\left(a-1\right)^2}{\left(2\sqrt{a}\right)^2}\cdot\dfrac{-4\sqrt{a}}{a-1}\)
\(=\dfrac{\left(a-1\right)^2}{4a}\cdot\dfrac{-4\sqrt{a}}{a-1}\)
\(=\dfrac{-\left(a-1\right)}{\sqrt{a}}\)
\(=\dfrac{1-a}{\sqrt{a}}\)
