\(\left(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{x}\right)^2\)
\(=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}+2\left(\dfrac{1}{xy}+\dfrac{1}{xz}+\dfrac{1}{yz}\right)\)
\(=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}+2\left(\dfrac{z}{xyz}+\dfrac{y}{xyz}+\dfrac{x}{xyz}\right)\)
\(=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}+2\left(\dfrac{x+y+z}{xyz}\right)\)
\(=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}+2\left(\dfrac{0}{xyz}\right)\)
\(=\dfrac{1}{x^2}+\dfrac{1}{y^2}+\dfrac{1}{z^2}\)