= Solution
{c}
Assume an initially unexcited <chemical atom>, two nonoverlapping square pulses starting at $0,T$, and a common coherent field phase. In the weak-excitation limit $|\Omega|\tau_p\ll1$, substitute $c_1=1+O(\Omega^2)$ into the second <probability amplitude> equation. After the second pulse,
$$
\begin{aligned}
c_2&=-\frac{i\Omega}{2}\left(\int_0^{\tau_p}e^{-i\delta t}\,dt+\int_T^{T+\tau_p}e^{-i\delta t}\,dt\right)+O(\Omega^3)\\
&=-i\Omega\tau_p\,e^{-i\delta(T+\tau_p)/2}\operatorname{sinc}(\delta\tau_p/2)\cos(\delta T/2)+O(\Omega^3),
\end{aligned}
$$
where the <sinc function> means $\operatorname{sinc}u=\sin u/u$, with value one at zero. Squaring gives
$$
\boxed{P_2=(\Omega\tau_p)^2\operatorname{sinc}^2(\delta\tau_p/2)\cos^2(\delta T/2)+O((\Omega\tau_p)^4).}
$$
The printed prefactor is a factor of four too small for the supplied equations and pulse durations. A direct counterexample is $\delta=0$: the exact resonant <Rabi oscillation> gives $P_2=\sin^2(\Omega\tau_p)$ for two pulses of duration $\tau_p$, because the intervening resonant rotating-frame evolution is trivial. Its small-area limit is $(\Omega\tau_p)^2$, rather than $(\Omega\tau_p/2)^2$. The latter is the single-pulse result before coherent addition.
The condition $\omega\tau_p\ll1$ printed in the source also does not justify the derivation from a physical carrier field. A consistent <rotating-wave approximation> pulse regime is
$$
\boxed{\omega_0^{-1}\ll\tau_p\ll|\Omega|^{-1},\qquad |\delta|\ll\omega_0.}
$$
If the supplied equations are treated as the mathematical model regardless, their weak-excitation integral remains valid for any $\tau_p$; subcycle pulses of the original real field require the discarded counter-rotating term and depend on carrier phase. $T$ here is the separation of pulse starts, as specified, rather than the dark interval $T-\tau_p$.
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