= Monotone parametrization of sine perturbation roots
{title2=$E'(x)=-2[x\cos(x/2)-\sin(x/2)]^2/(x^2-1)^2$}
The real roots of $\sin x=x+\epsilon(1-x^2)$, with $\epsilon>0$, can be counted by solving for the parameter: $\epsilon=E(x)=(x-\sin x)/(x^2-1)$. Positive values occur only on $(-1,0)$ and $(1,\infty)$. Direct <differentiation> and the half-angle identities give the derivative in the title. It is negative except at isolated points, so $E$ is strictly decreasing on both intervals, with limits $+\infty$ and zero. There are exactly two real roots for each positive parameter. As $\epsilon\to0^+$, <dominant balance> gives $x_-\sim-(6\epsilon)^{1/3}$ and $x_+\sim\epsilon^{-1}$. The exact root count supplements the local <asymptotic expansions>: a small-root expansion alone would omit the root escaping to infinity.
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