Verify 4.15 Ibach and Luth

Block[{system, f, m1, omegaSq, m2, q, a, d},
system = {
{ 2 f / m1 - omegaSq, -f (1 + E^(-I q a))/Sqrt[m1 m2]},
{  -f (1 + E^(I q a))/Sqrt[m1 m2], 2 f / m2 - omegaSq} 
 } ;

d = Det[ system ]  // FullSimplify ;
{Solve[ 0 == d, omegaSq] /. Cos[ a q ] -> 1 - 2Sin[ a q/2]^2
, system // MatrixForm } /. omegaSq -> ω^2
]

(* check *)
1 - 2Sin[ a q/2]^2  // Simplify