Is it possible that NSolve gives the roots that the function is not (well-)defined at those points? (an example is given)

Question

I have this function for reals values of $x$ for $2<x<3$. $$f(x)=(3 \pi -2 x)^2 \sin \left(\frac{1}{2} \left(\csc ^{-1}\left(\frac{4 \pi (\pi -x) \csc \left(\frac{\pi ^2}{\pi -x}\right)}{4 x^2-8 \pi x+5 \pi ^2}\right)-\frac{\pi x}{\pi -x}+\pi \right)\right)\\-(\pi -2 x)^2 \sin \left(\frac{1}{2} \left(\csc ^{-1}\left(\frac{4 \pi (\pi -x) \csc \left(\frac{\pi ^2}{\pi -x}\right)}{4 x^2-8 \pi x+5 \pi ^2}\right)+\frac{\pi ^2}{\pi -x}\right)\right)$$

I try NSolve to obtain all the roots of the function over the domain

f = -(\[Pi] - 2 x)^2 Sin[ 1/2 (\[Pi]^2/(\[Pi] - x) +   ArcCsc[(4 \[Pi] (\[Pi] - x) Csc[\[Pi]^2/(\[Pi] - x)])/( 5 \[Pi]^2 - 8 \[Pi] x + 4 x^2)])] + (3 \[Pi] - 2 x)^2 Sin[ 1/2 (\[Pi] - (\[Pi] x)/(\[Pi] - x) +  ArcCsc[(4 \[Pi] (\[Pi] - x) Csc[\[Pi]^2/(\[Pi] - x)])/( 5 \[Pi]^2 - 8 \[Pi] x + 4x^2)])];
NSolve[f == 0 && 2 < x < 3, x, Reals]
({{x -> 2.35619}, {x -> 2.61799}, {x -> 2.74889}, {x -> 2.82743}, {x ->
    2.87979}, {x -> 2.91719}, {x -> 2.94524}, {x -> 2.96706}, {x -> 
   2.98451}, {x -> 2.99879}})

Here, I see that all the roots are rational multiples of $\pi$

Rationalize[{1.5707963267948954`, 2.356194490192345`, 
   2.6179938779914944`, 2.748893571891069`, 2.827433388230814`, 
   2.879793265790644`, 2.9171931783333793`, 2.9452431127404313`, 
   2.9670597283903604`, 2.9845130209103035`, 2.99879298751753`}/Pi]
({1/2, 3/4, 5/6, 7/8, 9/10, 11/12, 13/14, 15/16, 17/18, 19/20, 21/22})

On the other hand, checking the function at these values, we see that the arguments of $\csc \left(\frac{\pi ^2}{\pi -x}\right)$ at these values are $\frac{\pi ^2}{\pi -x}=\{2 \pi ,4 \pi ,6 \pi ,8 \pi ,10 \pi ,12 \pi ,14 \pi ,16 \pi ,18 \pi ,20 \pi ,22 \pi \}$, accordingly, the function is not defined at these points. Then, should I ignore these soultions for $x$? Or, I am going wrong somewhere.

Any comments are appreciated.