Similarities between the trigonometric function and the lemniscate function from arithmetic view point (Q819267)

Let \(\text{sl }z\) and \(\text{cl }z\) denote the lemniscatic sine and cosine; for elements \(\mu \in \mathbb Z[i]\) with odd norm, consider the functions \(R_{s,\mu}(\text{sl }z) = \frac{\text{sl }\mu z}{\text{sl }z}\) and \(R_{c,\mu}(\text{sl }z) = \frac{\text{cl }\mu z}{\text{cl }z}\). In his analytic proofs of quartic and octic reciprocity laws, Eisenstein showed that \(R_{s,\mu}(\text{sl}\;z)\) is a quotient of two polynomials in \((\text{sl}\;z)^4\). In this article, the author shows that \(R_{s,\mu}(x) = \frac{K_\mu(x^2)}{K_\mu(-x^2)}\) for some polynomial \(K_\mu(x) \in \mathbb Z[i]\), and that \(R_{s,\mu}\big(\sqrt{\frac{1-x^2}{1+x^2}}\,\big) = R_{c,\mu}(x)\). Analogous result for the ordinary sine and cosine function are also derived.