= Solution
Put $r=B/A$. Equal two-body <phase space> gives $\Gamma_{S,L}\propto|A|^2|1\pm r|^2/2$, so the <decay-rate asymmetry> is exactly
$$
\mathcal R=\frac{|1+r|^2-|1-r|^2}{|1+r|^2+|1-r|^2}
=\frac{2\operatorname{Re}r}{1+|r|^2}.
$$
The stated magnitude assumption fixes $|r|=\tan^2\theta_C$. Retaining the relative <strong phase> $\delta$ in the convention of <2/b> therefore gives
$$
\boxed{\mathcal R=\frac{2\tan^2\theta_C\cos\delta}{1+\tan^4\theta_C}
=2\tan^2\theta_C\cos\delta+O(\tan^6\theta_C).}
$$
\b[A magnitude assumption alone does not determine $\mathcal R$.] For example, $r=i\tan^2\theta_C$ has the stipulated magnitude but gives $\mathcal R=0$, whereas a real positive $r$ gives a positive asymmetry. With the additional aligned-amplitude assumption $\delta=0$, conventionally used for this idealised extraction, the intended leading result is $\boxed{\mathcal R\simeq2\tan^2\theta_C}$. The sign of $r$ here refers to the complete amplitude after choosing the <kaon> phase convention, not to a bare <CKM matrix> product. A flavour-symmetry relation can supply further information, but is stronger than the printed magnitude assumption.
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