= Solution
Assume a uniform stellar disc of radius $R_\star$, an opaque planetary radius $R_p$, and a thin atmospheric annulus of thickness $z\ll R_p$. Neglect planetary light during transit and scattering back into the beam. The attenuation part of the <radiative transfer equation> gives $I_\lambda=I_{\lambda,0}e^{-\tau_\lambda}$, where $\tau_\lambda$ is the slant <optical depth> along a stellar ray through the atmosphere. The fraction of light removed from that annulus is therefore $1-e^{-\tau_\lambda}$.
If the annulus is represented by one effective <optical depth>, its area divided by the stellar area is
$$
A=\frac{\pi[(R_p+z)^2-R_p^2]}{\pi R_\star^2}\simeq\frac{2R_pz}{R_\star^2}.
$$
The <annulus model for transmission spectroscopy> consequently gives the extra transit depth
$$
\boxed{\delta_\lambda=A(1-e^{-\tau_\lambda}).}
$$
Here $\delta_\lambda$ is normalized to the unobscured stellar flux and excludes the opaque-disc depth $(R_p/R_\star)^2$. For a spectral feature measured relative to a continuum with slant <optical depth> $\tau_c$, the corresponding contrast is $A(e^{-\tau_c}-e^{-\tau_\lambda})$; the displayed formula takes the annular continuum to be transparent.
A real <exoplanet transmission spectrum> has an impact-parameter-dependent <optical depth>. Its more accurate expression is
$$
\delta_\lambda=\frac2{R_\star^2}\int_{R_p}^{\infty}b\,[1-e^{-\tau_\lambda(b)}],db.
$$
Stellar limb darkening and horizontally varying clouds further modify the weighting. The single-annulus approximation is useful for estimating an <atmospheric spectral-feature amplitude>, rather than predicting every spectral line.
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