A solution  $(x,y)$of the system of equations  $x-y=\frac{1}{3}$and  ${\cos }^2\left(\pi x\right)-{\sin }^2\left(\pi y\right)=\frac{1}{2}$ given by

$x-y=\frac{1}{3};{\cos }^2\pi x-{\sin }^2\pi y=\frac{1}{2}$

$\cos \pi (x+y)\cos \left(\pi \left(\frac{1}{3}\right)\right)=\frac{1}{2}$

$\cos \pi (x+y)=2n\pi =1$

$\therefore x+y=2$

$x-y=\frac{1}{6}$

$\therefore x=2+\frac{1}{6},y=2-\frac{1}{6}$

$(x,y)=\left(\frac{13}{6},\frac{11}{6}\right)$