\(\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}+\dfrac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\)
\(=\dfrac{\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)-\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)+xy-1}{xy-1}\)
\(=\dfrac{x\sqrt{y}-\sqrt{x}+\sqrt{xy}-1-xy-\sqrt{xy}-x\sqrt{y}-\sqrt{x}+xy-1}{xy-1}\)
\(=\dfrac{-2\sqrt{x}-2}{xy-1}\)
\(1-\dfrac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}\)
\(=\dfrac{xy-1-\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)-\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)}{xy-1}\)
\(=\dfrac{xy-1-xy-\sqrt{xy}-x\sqrt{y}-\sqrt{x}-x\sqrt{y}+\sqrt{x}-\sqrt{xy}+1}{xy-1}\)
\(=\dfrac{-2\sqrt{xy}-2x\sqrt{y}}{xy-1}\)
Ta có: \(\left(\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}+\dfrac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):1-\left(\dfrac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\dfrac{\sqrt{x}+1}{\sqrt{xy}+1}\right)\)
\(=\dfrac{-2\left(\sqrt{x}+1\right)}{xy-1}:\dfrac{-2\sqrt{xy}\left(\sqrt{x}+1\right)}{xy-1}\)
\(=\dfrac{-2\left(\sqrt{x}+1\right)}{xy-1}\cdot\dfrac{xy-1}{-2\sqrt{xy}\left(\sqrt{x}+1\right)}=\dfrac{1}{\sqrt{xy}}\)
