Norman Wildberger's "rational trigonometry" has been viewed by some mathematicians as a clever new take on an ancient topic. Wildberger's "spread polynomials" $S_n$ are characterized by the identity $$ \sin^2(n\theta) = S_n(\sin^2\theta) $$ (except that Wildberger refuses to refer explicitly to the sine function in the definition and does it by other means). In one sense these are trivially equivalent to the Chebyshev polynomials $T_n$ characterized by $$ \cos(n\theta) = T_n(\cos\theta). $$ Wildberger notes that $1 - 2S_n(s) = T_n(1 - 2s)$.

In thinking about whether this polynomial sequence is even worth mentioning after Chebyshev polynomials have been treated, three questions come to mind:

Could it be that an essential difference justifying a separate treatment is the factorization of these polynomials? Wildberger factors the spread polynomials. Is there some important reason for doing that?

Is there a combinatorial interpretation of the coefficients? The coefficient of the $n$th-degree term in $S_n$ for $n=1,\dots,10$ is $n^2$ and the coefficient of the constant term is $4^{n-1}$. The first-degree term is $n/2$ times the constant term.

If there's no essential difference that justifies attending to the two sequences separately, the fact that the conventional way of viewing the sequence (Chebyshev polynomials) is chronologically first, doesn't mean it's necessarily better than Wildberger's way of viewing it (spread polynomials). Is it?