When I learnt these (about 200 years ago) I was recommended to use words, not symbols. After all, that's the way most of us learn our tables. It's a question of establishing a rhythm in your head. I've always recommended my students to learn them this way.
Originally Posted by transgalactic
It goes like this:
Sine plus sine is twice sine half-sum, cosine half-difference.The next one swaps the sine and cosine around on the RHS:
Sine minus sine is twice cos half-sum, sine half-difference.The next is the line of all cosines:
Cos plus cos is twice cos half-sum, cos half-difference.The last one you remember, simply because it's different from all the rest; plus, being cosines it's the wrong way round. (Cosine is always awkward - when you differentiate it, for instance, you get a minus sine - pun intended.)
Cos minus cos is twice sine half-sum, sine half-difference reversed.Just repeat these words over and over - you'll end up remembering them, just like learning a poem. Take my advice - don't try to learn them by writing them down using symbols and letters.