The addition formulas for sine and cosine look like this:

I can never remember them.

One solution is of course to look them up in a book or search the internet. What I usually do, though, is derive them using complex arithmetic. Apart from the usual rules known from real-valued arithmetic, all that is needed is knowing and . Then you get

By equating the real and imaginary parts you get the answer.

Flicking through *Proofs Without Words II* by Roger B. Nelsen, I saw the following wonderful figure. It could be a contender to an easier way to remember the addition formulas.

(Attributed to the author himself.) It should be pretty much self-explanatory. Apart from using sine and cosine to assign side-lengths to the four relevant right-angled triangles, all you need know is that the sum of the angles in a triangle is equal to two right angles (to realize that the two -angles are indeed equal).

Both volumes of *Proofs Without Words* contain other “visual proofs” of the addition formulas. Some of these can also be found online.

How do you remember the addition formulas for sine and cosine?