The points are on the graphs x=sqrt(1+2cos(y)) and x=-sqrt(1+2cos(y)) which form a vertical arrangement of ovals.  (blue and green on the graph)

The new function can then be split into

f(x)=sqrt(1+2cos(x))-cos(x)

g(x)=-sqrt(1+2cos(x))-cos(x)

(Which form the horizontal red and black heart shapes on the graph.)

https://www.desmos.com/calculator/vdhdfeewga

I'll spare the formulas, but using the derivative: f'(x)=0 when sin(x)=0 or cos(x)=0 from which the maximum is f(pi/2)=1

g'(x)=0 when sin(x)=0 from which the minimum is f(0)=-1-sqrt(3)