All these identities follow a pattern. If we think of i, -1, -i, 1 as the four "signs" each function can have, then we can try different combos of sine and cosine with different signs between them and see what pops out. To get all permutations, take these basic four and multiply by powers of i on both sides.
At the top we have linear combos, circular on the left, hyperbolic on the right.
After that are all the quadratic combos with two terms. Again, to get the rest, multiply both sides by powers of i.
I didn't include things like (sine + sine) or (sine^2 + sine^2) cuz well, theyre just not interesting enough. I also didn't include (sine + i*sine) cuz that shows up later on the list with an extra factor of cosine.