First, a side question. I was going through the derivation found at:
https://www2.ph.ed.ac.uk/~mevans/em/lec5.pdf
My first question: Where does the cos(theta) come from in the equation just below (1)? $$\frac{a}{r}{cos\theta}$$
Main question. In this derivation of the charge-dipole interaction, starting on page 6:
A Taylor Expansion is used to determine how the electric field behaves at x >> a. My question is about Taylor Expansions in general. Looking at the three equations (8) on page 6, I would never have guessed or thought to use a Taylor Expansion. I know how to use the Taylor Expansion for the typical equations (i.e. $e^x$, $sin(x)$, etc...), but I really struggle with recognizing when to use it. This derivation is a perfect example. What in this equation would give me a clue to use the Taylor Expansion? Why does an expansion need to be used here?
$cos(\theta)$ is the angle that comes from the product $a\cdot r$.
About Taylor expansions, it's more a matter of practice. In general, when you have an expression in the form $(X)^-k$, and $X$ is your variable, you want to do a low-order approximation if possible. And sometimes, Taylor expansion can help to solve integral.
Here, you have $(1-a/x)^-2$ which is kind of hard to get a behaviour, so you have to do an expansion.
Once again, it's more a matter of practice and experience, it's part of the things that come with time.