calculus and analysis – Why that difference in outputs of FunctionInjective?

Trying a new in 12.2 command FunctionInjective, I execute

FunctionInjective({Sin(x)^2, Element(x, Integers)}, x)

False

and

FunctionInjective({Cos(x)^2, Element(x, Integers)}, x)

True

Both function are even on the reals, so both outputs should be False as the output of

DiscretePlot(Cos(x)^2, {x, -8, 8})

enter image description here

shows,

How to explain this difference?

I try to present Cos(x)^2 as Sin(Pi/2 - x)^2 and (1+Cos(2x))/2, but
the outputs of

FunctionInjective({Sin(Pi/2 - x)^2, Element(x, Integers)}, x)

and

FunctionInjective({(Cos(2*x) + 1)/2, Element(x, Integers)}, x)

still are True.