The ratio of the greatest value of $2-\cos x+{\sin }^{2}x$ to its least value is

$2-\cos x+{\sin }^{2}x=-({\cos }^{2}x+\cos x)+3$

$=-\left({\left(\cos x+\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)}^2-\frac{{\displaystyle 1}}{{\displaystyle 4}}\right)+3=\frac{13}{4}-{\left(\cos x+\frac{{\displaystyle 1}}{{\displaystyle 2}}\right)}^2$

maximum value is at $\cos x=-\frac{1}{2}$ and

maximum value $=\frac{13}{4}$ minimum value is at 

$\cos x=1$ and minimum value

$=1,$ ratio $=\frac{13}{4}=13:4$