<s>[INST] How to refactor $\pi / 12$ into $\pi /4 - \pi /3$? How to refactor $\pi / 12$ into $\pi /4 - \pi /3$ (standard calculus angles)? How do you know that $\pi / 12$ converts exactly into $\pi /4 - \pi /3$? Do I need to memorize that or is there a simple procedure to follow in order to refactor any angle into combination of standard calculus angles ($\pi /4, \pi /3, \pi /6$) for which sin/cos values are well known?
Similar for $7\pi / 12 = \pi /4 + \pi /3$
 [/INST] A good place to start is by expressing each of the standard angles as multiples of $\frac{\pi}{12}$.
\begin{align*}
\frac{\pi}{6} & = \frac{2\pi}{12}\\
\frac{\pi}{4} & = \frac{3\pi}{12}\\
\frac{\pi}{3} & = \frac{4\pi}{12}
\end{align*}
From there, you can immediately see that 
\begin{align*}
\frac{\pi}{12} & = \frac{\pi}{4} - \frac{\pi}{6}\\
               & = \frac{\pi}{3} - \frac{\pi}{4}
\end{align*}
and that 
\begin{align*}
\frac{5\pi}{12} & = \frac{\pi}{4} + \frac{\pi}{6}\\
\frac{7\pi}{12} & = \frac{\pi}{3} + \frac{\pi}{4}
\end{align*}
For a larger angle such as $17\pi/12$, look for linear combinations of the special angles.  For instance, 
\begin{align*}
\frac{17\pi}{12} & = \frac{9\pi}{12} + \frac{8\pi}{12}\\
                 & = \frac{3\pi}{4} + \frac{2\pi}{3}
\end{align*}
It is less useful to express $17$ as $10 + 7$ since $7$ is not a multiple of $2$, $3$, or $4$.  Similarly, it is not helpful to express $17$ as $11 + 6$, $12 + 5$, $13 + 4$, $14 + 3$, or $16 + 1$ since one of the summands is not a multiple of $2$, $3$, or $4$.  On the other hand, we can express $17$ as $15 + 2$, which yields
\begin{align*}
\frac{17\pi}{12} & = \frac{15\pi}{12} + \frac{2\pi}{12}\\
                 & = \frac{5\pi}{4} + \frac{\pi}{6}
\end{align*}
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