Exact image of the p-adic cosine series on its maximal domain

Characterize the exact image cos(p^d Z_p) ⊂ Q_p of the p-adic cosine series cos(x)=∑_{i=0}^∞ ((−1)^i x^{2i})/(2i)! restricted to its domain p^d Z_p (with d=2 if p=2 and d=1 otherwise). Determine the precise subset of Q_p that this image equals, rather than a mere inclusion.

Background

The paper establishes that the cosine series converges on p^d Z_p and proves the image is contained in 1 + p^{2d−1} Z_p, but does not determine the exact image set. In contrast, exact image characterizations are provided for the p-adic exponential and sine series.

A full description of cos(p^d Z_p) would complete the trigonometric series analysis and strengthen several constructions that rely on understanding p-adic rotations and their parameterizations.