I notices something more:

If I choose omega=0 therefore Table[test[0, s[i], t[i], u[i]], {i, 1, 5}] he gives me outputs that such as True,False

But if omega is anything else then 0,Pi/2 and I run:

test[omega_, psiFix_, phiFix_, delta_] := 
 FullSimplify[-Cos[omega] Cos[psi] Sin[phi] + Sin[omega] Cos[phi] >= 
   0, psiFix - delta < psi < psiFix + delta && 
   phiFix - delta < phi < phiFix + delta]

Table[test[0.1, s[i], t[i], u[i]], {i, 1, 5}]

I get such crappy outputs:

{1. Cos[phi] >= 9.96664 Cos[psi] Sin[phi], 
 1. Cos[phi] >= 9.96664 Cos[psi] Sin[phi], 
 1. Cos[phi] >= 9.96664 Cos[psi] Sin[phi], 
 1. Cos[phi] >= 9.96664 Cos[psi] Sin[phi], 
 1. Cos[phi] >= 9.96664 Cos[psi] Sin[phi]}