= Solution
Work in the source rest frame and neglect the source's small <cosmological redshift>. Over an emission interval $\Delta t$, a ballistic <relativistic jet> feature travels $\beta c\Delta t$, advancing $\beta c\Delta t\cos\theta$ toward the observer and $\beta c\Delta t\sin\theta$ across the sky. The second <photon> has a shorter journey, so its arrival is delayed by only
$$
\Delta t_{\rm obs}=\Delta t(1-\beta\cos\theta).
$$
Dividing the projected displacement by this arrival interval gives
$$
\boxed{\beta_{\rm app}=\frac{\beta\sin\theta}{1-\beta\cos\theta}.}
$$
<Apparent superluminal motion> is a combination of projection and light-travel delay. The feature's local <speed> remains $\beta c<c$; its inferred transverse displacement divided by the observer's arrival interval is not a local signal <velocity>.
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