Well, we can use mathematical analysis.
If we want to find
$\min$ of
$-2 \cdot f$ then we need to find
$\max$ of
$f$.
So it is root of first derivative with negative second derivative.
I rewrote your expression a little for simplicity, so we try this:
Clear[a, b, t, V];
V[x_] := a Cos[x + t/2] + b Cos[x - t/2];
And if we run next code, in 15-20 sec (on average laptop) we get big dump,
but with meaningful and useful results:
goal = D[V[x], x];
secDeriv = D[V[x], {x, 2}];
Reduce[goal == 0 && secDeriv < 0 && 0 <= x <= 2 \[Pi], x, Reals]
