1. Equation of motion of a forced oscillator

For a mass-spring system without damping, the equation is

$m\ddot x + kx = F(t).$

Since the natural angular frequency is

$\omega_0 = \sqrt{\frac{k}{m}},$

we have

$k = m\omega_0^2.$

So the equation becomes

$m\ddot x + m\omega_0^2 x = F(t).$

2. Form of the driving force

The external force is proportional to $\cos \omega t$, so let

$F(t) = F_0 \cos \omega t.$

We look for the steady-state displacement in the form

$x(t) = A\cos \omega t.$

Then

$\ddot x = -A\omega^2 \cos \omega t.$

Substitute into the equation of motion:

$m(-A\omega^2 \cos \omega t) + m\omega_0^2(A\cos \omega t) = F_0\cos \omega t.$

Factor out $A\cos \omega t$:

$m(\omega_0^2 - \omega^2)A\cos \omega t = F_0\cos \omega t.$

Hence,

$A = \frac{F_0}{m(\omega_0^2 - \omega^2)}.$

Therefore, the displacement is proportional to

$\frac{1}{m(\omega_0^2 - \omega^2)}.$

3. Match with options

This matches:

$\boxed{\text{B: } \frac{1}{m(\omega_0^2 - \omega^2)}}$

4. Comparison with stored correct answer

Stored correct answer: B

Our derived answer: B

So they agree.