= Solution
Use the integrated constant-compressibility model from <1/c>, with $\kappa p=0.0059\times100=0.59$:
$$
a(100)=0.287e^{-0.59/3}\ \mathrm{nm}=0.235760\ \mathrm{nm},\qquad
\sin\theta(100)=0.37942\ldots e^{0.59/3}.
$$
Therefore
$$
\boxed{2\theta(100\ \mathrm{GPa})=55.018^\circ.}
$$
Compression reduces the spacing and moves the same allowed reflection to larger deflection, as the positive <pressure> <derivative> predicts. The input <Isothermal compressibility>[compressibility] corresponds to a constant <bulk modulus> $1/\kappa\simeq169.5$ GPa in this ideal model.
This is the stipulated \b[unchanged cubic-phase, constant-compressibility extrapolation], rather than a reliable prediction for real Fe at 100 GPa. Real Fe exhibits a pressure-induced <Body-centered cubic lattice>[body-centered-cubic] to <Hexagonal close-packed structure>[hexagonal-close-packed] transformation, and <Isothermal compressibility>[compressibility] can change with <pressure>.https://journals.aps.org/prb/abstract/10.1103/PhysRevB.78.220101[2008 primary diffraction study, abstract: pressure-driven structural transformation in Fe]{ref} A real high-pressure <X-ray diffraction>[diffraction] interpretation would require its actual phase and <equation of state>. The source gives neither, so its numerical exercise is answered within its explicitly stated idealization.
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