= Solution
Apply the <product-to-sum formula> $2\cos u\cos v=\cos(u+v)+\cos(u-v)$ to the <amplitude modulation>:
$$
\boxed{A=a\cos(\omega t-kx)
+\frac b2\cos\bigl((\omega+\omega_m)t-kx\bigr)
+\frac b2\cos\bigl((\omega-\omega_m)t-kx\bigr).}
$$
The <amplitude modulation sidebands> have <angular frequencies> $\omega+\omega_m$ and $\omega-\omega_m$, each with <amplitude> $b/2$ in this real <cosine> convention. The <carrier wave> has <angular frequency> $\omega$ and <amplitude> $a$.
This identity is exact for the supplied space-time expression. A physical qualification is needed if every term is meant to be a freely propagating <electromagnetic wave> with <speed of light> $c$: the <dispersion relation> requires $k_\pm=(\omega\pm\omega_m)/c$, whereas the prescribed modulation at every position leaves all three <wavenumbers> equal to $k=\omega/c$. A freely propagating modulated <wave> instead has envelope $a+b\cos(\omega_m(t-x/c))$; its analogous expansion has $k_\pm$. For example, at $x=t=0$ the prescribed expression has $A_{tt}-c^2A_{xx}=-b\omega_m^2$, which is nonzero for nontrivial modulation. The given formula can describe modulation imposed throughout space or the temporal signal at one position.
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