Completion of the elementary-function chain for the cosine-generated operator

Complete the elementary-function reconstruction chain for the Stachowiak operator generated by $f(x)=\cos x$, including the still-unreached reciprocal and division operations.

Background

The paper describes Stachowiak’s general construction and notes that the example $f(x)=\cos x$ can be used to obtain multiplication with two distinguished constants through prosthaphaeresis. However, this partial result does not provide the full set of elementary operations.

The authors explicitly report that the complete elementary-function chain was not obtained and that reciprocal and division remained unreachable in their treatment.