Then it occurred to me that there is a fairly obvious point about this that should make it even clearer. Just remember that pattern is sine sine, cosine cosine, both terms positive.

Then suppose

`s = t`

, we haveSo, which function and for what combination of s with itself would we always get 1? Well, it's obviously the difference, which always equals zero (the sum, 2s, could be any angle). And which function always gives 1 with an argument of 0? The cosine, of course.

Getting to the formula for

`cos(s+t)`

just involves realizing that if we plug in `u = -t`

we havebut

So it's the sine term in the formula that changes sign when we add.

As for the other one, perhaps the easiest is Euler:

The real part gives us what we had before,

and the imaginary part is equal to the imaginary part of the sum from the previous line:

In fact, maybe this is enough by itself. :)