KZ-MO 2017 Problem 2

For positive reals $x,y,z\ge \frac{1}{2}$ with $x^2+y^2+z^2=1$ , prove this inequality holds $$(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2$$