My opinion is that there is no need to remember the formula for sine angle addition, that is, that $\sin(a)\cos(b)+\sin(b)\cos(a) = \sin(a+b)$. My reasoning is as follows:
Case 1: If you have both angles $a$ and $b$, simply calculate $\sin(a+b)$ directly.
Case 2: Let's say you start off with, not the angles, but the actual measurements themselves, to which the sine and cosine of a will be $y_1$ and $x_1$, and the sine and cosine of $b$ will be $y_2$ and $x_2$. From here, we can calculate angles $a$ and $b$ via $\arctan(y_1/x_1)$ and $\arctan(y_2/x_2)$, respectively, and then just do what we did previously in case 1.
Additionally, if you have one angle, but you only have the $x/y$ measurements of the other angle, then this is really just case 2 again.
Thus, the sine angle addition formula isn't necessary in order to advance ones knowledge of trigonometry, and can be safely done away with (if my reasoning is correct).
With this, is it safe to assume that I can file the sine angle addition formula under the "nice to know" category? Or is there something right under my nose which I've blindly missed? Honestly, this could very well be the case, hence the reason for this post to begin with.