= Solution
Use an orientation-preserving isometry to send $P$ to $i$ and $\ell$ to the imaginary axis, whose boundary points are $0,\infty$. A geodesic $m$ through $i$ meeting it at angle $\theta$ has boundary points
$$
u=-\tan\frac\theta2,
\qquad
v=\cot\frac\theta2.
$$
Indeed $uv=-1$, so the semicircle with endpoints $u,v$ passes through $i$, and its tangent there makes angle $\theta$ with the vertical.
The <cross-ratio> is invariant under the normalizing Möbius transformation. With the suitable ordering $(u,v,\infty,0)$ and the convention used here,
$$
[u,v,\infty,0]
=\frac{v}{v-u}
=\frac{\cot(\theta/2)}
{\cot(\theta/2)+\tan(\theta/2)}
=\boxed{\cos^2\frac\theta2}.
$$
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