I'm reading this part of Landau, Lifshitz, The Classical Theory Of Fields:
I'm able to derive the unnumbered formula for $\sin\theta-\sin\theta^\prime$, finding expansion to Taylor series, but what I fail to understand is how they derived $(5.7)$. I can prove this result in the limit of $\Delta\theta\to0$ by dividing unnumbered formula by $(5.7)$ and prove that limits of both parts are identical, but how to derive it?
Also, should one understand "to the same order of accuracy" as for $\Delta\theta^\prime\to0$ not for $\frac Vc\to0$?
For the sake of convenience, set $\Delta \theta' = - \Delta \theta$. You can always write $\theta = \theta' + \Delta \theta'$. Substituting in unnumbered equation
\begin{equation} \sin(\theta' + \Delta \theta' ) - \sin\theta' = -\frac{V}{c} \sin \theta' \cos \theta' \end{equation}
Now divide and multiply by $\Delta \theta'$ and take the limit $\Delta \theta' \to 0$. So
\begin{equation} \lim_{\Delta \theta' \to 0} \Delta \theta' \frac{\sin(\theta' + \Delta \theta' ) - \sin\theta'}{\Delta \theta'} = \Delta \theta' \cos\theta' = -\frac{V}{c} \sin \theta' \cos \theta' \end{equation}
Hence, now substituting $\Delta \theta' = -\Delta \theta$ you gain the result.