= Solution
In charge-free, current-free vacuum, <Maxwell's equations> are
$$
\nabla\cdot\mathbf E=0,\qquad \nabla\cdot\mathbf B=0,\qquad
\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t},\qquad
\nabla\times\mathbf B=\mu_0\epsilon_0\frac{\partial\mathbf E}{\partial t}.
$$
Taking the <curl> of the third equation and using the fourth, the <curl of the curl identity>, and the zero <divergence> of the <electric field> gives
$$
\nabla^2\mathbf E=\mu_0\epsilon_0\frac{\partial^2\mathbf E}{\partial t^2}.
$$
The <magnetic flux density> obeys the same <wave equation>. For the <plane wave> factor $e^{i(\omega t-kx)}$, the spatial <Laplacian> contributes $-k^2$ and the second time <derivative> contributes $-\omega^2$. Thus
$$
k^2=\mu_0\epsilon_0\omega^2,\qquad
\boxed{c=\frac{\omega}{k}=\frac1{\sqrt{\mu_0\epsilon_0}}}
$$
for propagation in the positive $x$ direction. The zero-<divergence> conditions require $E_{0x}=B_{0x}=0$, while the <curl> equations require $\mathbf B_0=(k/\omega)\hat{\mathbf x}\times\mathbf E_0$. These conditions show that the proposed <plane waves> satisfy all four <Maxwell's equations>, rather than only their derived <wave equations>. Physical <electric fields> and <magnetic fields> are the <real parts> of the complex expressions.
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