Here is how to get the compound angle formulas (including double/half/sum/difference……).
[Since you are only looking for a way to help you to memorize the result, the development involved will not be backed by reasons.]
Among all these formulas, only p: sin (A + B) = sin A cos B + cos A sin B needs to be memorized and all the others are just its “derivatives”.
Putting B = –b in (p) to get p1: sin (A – b)...............(*)
Differentiating (p) [wrt to A only] to get q : cos (A + B)
Same as (*) to get r : cos(A – B)
s: tan (A + B) is just (p) / (q). And tan (A – B) can be obtained similarly.
By letting A = B in (p), (q) and (s) to obtain the corresponding double angle formulas. In particular, $t_1: cos 2A = 1 – 2 sin^2 A$ and $t_2: cos 2A = 2 cos^2 A – 1$.
By re-arranging terms in $t_1$ and $t_2$, formulas for $sin^2 A$ and $cos^2 A$ can then be obtained.
Performing (p) + (p1), we get the product-to-(sum & difference) formula. Other formulas can be obtained similarly.
Sum-to-product can also be done with a little imagination. Hope you can take it from here.