= Solution
For equal <complex amplitudes>, the <superposition principle> and the <complex exponential> formula for the cosine give the exact identity
$$
\begin{aligned}
y&=Ae^{i[(k_0+\delta k)x-(\omega_0+\delta\omega)t]}
+Ae^{i[(k_0-\delta k)x-(\omega_0-\delta\omega)t]}\\
&=\boxed{2A\cos(\delta k\,x-\delta\omega\,t)\,e^{i(k_0x-\omega_0t)}}.
\end{aligned}
$$
The cosine is a real periodic modulation of the rapidly varying carrier. It varies slowly when $|\delta k|\ll|k_0|$ and $|\delta\omega|\ll|\omega_0|$. Its signed <wave envelope> travels with velocity $\delta\omega/\delta k$; its magnitude gives the visible <amplitude> envelope.
Equal magnitudes need not mean equal initial <wave phases>. If the two <complex amplitudes> are $ae^{i\phi_+}$ and $ae^{i\phi_-}$, write $\bar\phi=(\phi_++\phi_-)/2$ and $\delta\phi=(\phi_+-\phi_-)/2$. The same result becomes $2ae^{i\bar\phi}\cos(\delta kx-\delta\omega t+\delta\phi)e^{i(k_0x-\omega_0t)}$, still with a real slowly varying modulation.
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