I'm struggling to understand the derivation of the equation of the centre given in Robin Green's Spherical Astronomy (p147-148).
To order $e^{2}$, the equation of the centre is $$\nu-M=2e\sin M+\frac{5}{4}e^{2}\sin2M,\tag{1}$$ where $e$ is orbital eccentricity, $\nu$ is the true anomaly and $M$ is the mean anomaly.
Green derives the first two terms as follows.
Starting with (this is derived earlier on - a derivation I can understand)$$M=\nu-2e\sin\nu+\frac{3}{4}e^{2}\sin2\nu+O\left(e^{3}\right)\tag{2}.$$Assume $\nu=M$ to zero order in $e$. Substitute this in the second rhs term of (2) to give$$\nu=M+2e\sin M+O\left(e^{2}\right).$$And rewrite (2) as$$\nu=M+2e\sin\left(M+2e\sin M\right)-\frac{3}{4}e^{2}\sin2M+O\left(e^{3}\right).$$
Green then goes on to say: “With the usual small-angle approximations, it is found that”$$\nu-M=2e\sin M+\frac{5}{4}e^{2}\sin2M+O(e^{3)}.$$
How does he make that small angle approximation?