Monotone parametrization of sine perturbation roots ($E^{\prime }(x)=-2[x\cos (x/2)-\sin (x/2){]}^2/(x^2-1{)}^2$)

The real roots of $\sin x=x+ϵ(1-x^2)$, with $ϵ>0$, can be counted by solving for the parameter: $ϵ=E(x)=(x-\sin x)/(x^2-1)$. 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 $ϵ\to 0^{+}$, dominant balance gives $x_{-}\sim -(6ϵ{)}^{1/3}$ and $x_{+}\sim {ϵ}^{-1}$. The exact root count supplements the local asymptotic expansions: a small-root expansion alone would omit the root escaping to infinity.