trigonometric inequation

i need to prove that for every $y\geq x\geq\frac{3}{\pi}$ : $y^{2}cos\frac{1}{y}-x^{2}cos\frac{1}{x}\geq(y^{2}-x^{2})cos\frac{1}{x}\geq\frac{(y^{2}-x^{2})}{2}$