In the book 'An Introduction to Modern Astrophysics' (Carroll and Ostlie, 2017), the first chapter presents an application of spherical trigonometry. At a certain point, the authors derive an equation (which I will omit here) that yields:
\begin{equation} \sin(\Delta \alpha) \cos(\delta + \Delta \delta) = \sin(\Delta \theta) \sin (\phi), \end{equation}
where $\delta$ represents the star's declination and $\alpha$ is the right ascension. $\Delta \theta$ corresponds to the angular distance traveled by the star from one point to another on the celestial sphere, while $\phi$ denotes the position angle calculated from the celestial north pole. In the text, the authors further state that, "Assuming that the changes in position are significantly smaller than one radian, we can employ the small angle approximations $\sin x \sim x$ and $\cos x \sim 1$. By employing the appropriate trigonometric identity and neglecting terms of second order or higher, the previous equation simplifies to":
\begin{equation} \Delta \alpha = \Delta \theta \frac{\sin \phi}{\cos \delta}. \end{equation}
However, this was my attempt:
\begin{align} \sin(\Delta \alpha) \cos(\delta + \Delta \delta) &= \sin(\Delta \theta) \sin\phi \nonumber\\ \sin(\Delta \alpha) \left[ \cos\delta\cos\left(\Delta \delta\right) - \sin\delta\sin\left(\Delta \delta\right) \right] &= \sin(\Delta \theta) \sin\phi \nonumber\\ \Delta \alpha \left[\cos\delta - \Delta \delta \sin \delta \right]&\approx \Delta \theta \sin\phi \nonumber\\ \Delta \alpha &= \frac{\Delta \theta \sin(\phi)}{\left[\cos\delta - \Delta \delta \sin \delta \right]} \end{align}
Why was it seemingly considered that $\Delta \delta \sin \delta \approx 0$? In subsequent derivations made by the authors, to find the change in declination using the law of cosines for sides, they do not consider, for example, $\cos \delta (\Delta \delta) \approx 0$, but rather as an arbitrary term to be calculated.
