= Solution
From <2/c>, $\sin(A/2)\ge1/\gamma$, with equality when $z=0$: the <photons> are emitted perpendicular to the <Lorentz boost> axis in the <center-of-momentum frame>. Therefore
$$
\boxed{A_{\min}=2\arcsin(\gamma^{-1}),\qquad A_{\mathrm{peak}}=A_{\min}.}
$$
At the minimum, the map $z\mapsto A$ is stationary: writing $A=A_{\min}+\delta$, its leading behavior is $\delta=\beta z^2/\gamma+O(z^4)$. A nonzero interval of uniformly distributed $z$ is therefore compressed near $A_{\min}$, producing an integrable <inverse-square-root singularity> in the opening-angle <probability density>. No <differentiation> of that <probability density> is needed. For finite $\gamma>1$, the <probability density> vanishes at $A=\pi$. For $\gamma=1$ the <center-of-momentum frame> is the laboratory and every event has $A=\pi$: the <probability distribution> becomes a <Dirac delta distribution>, rather than the singular $\beta=0$ substitution in <2/c>. At large $\gamma$, $A_{\min}\sim2/\gamma$ in radians.
\Image[/past-exam-of-the-symmetry-and-relativity-course-of-the-university-of-oxford/2022-diphoton-opening-angle.png]
{title=Opening-angle compression and its normalized probability density for a boosted two-photon decay}
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