= Solution
Choose the source motion along $+x$, write $\beta=v/c$, and let primes denote its <instantaneous rest frame>. A <photon> with rest-frame <energy> $E'$ and direction $\theta'$ has $p_x'=E'\cos\theta'/c$ and transverse momentum $E'\sin\theta'/c$. A <Lorentz boost> to the laboratory gives
$$
E=\gamma E'(1+\beta\cos\theta'),\qquad
p_x=\gamma\frac{E'}c(\cos\theta'+\beta).
$$
Since $|\mathbf p|=E/c$, the <relativistic aberration> relation and its inverse are
$$
\boxed{\cos\theta=\frac{\cos\theta'+\beta}{1+\beta\cos\theta'},\qquad
\cos\theta'=\frac{\cos\theta-\beta}{1-\beta\cos\theta}.}
$$
The mapping is monotone. The boundary of the rest-frame forward hemisphere, $\cos\theta'=0$, therefore maps to $\cos\theta=\beta$. An isotropic <photon> count assigns exactly half the <photons> to that hemisphere, proving
$$
\boxed{\theta_{1/2}=\cos^{-1}\beta=\sin^{-1}(1/\gamma).}
$$
The <photon count inside a relativistic beaming cone> concerns an emitted ensemble and counts <photons>, not their total <energy> or the arrival-time-weighted <power> at a detector. The laboratory <photon> energies depend on direction, so “half the <photons>” need not mean half the radiated <energy>. At large $\gamma$ the angular scale is $\theta_{1/2}\simeq1/\gamma$.
\Image[/past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2015-aberration-and-synchrotron-arrival-compression.png]
{title=Aberration of an isotropic photon count and compression of arrival times}
{description=Original evaluation of the angle map and the ultrarelativistic tangent-view arrival-time relation. The beaming panel uses an illustrative modest boost so the half-count cone is visible; the storage-ring beam has a much larger Lorentz factor.}
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