The addition formulas for sine and cosine look like this:

$\begin{aligned} \cos(\alpha + \beta) &= \cos \alpha \cos \beta - \sin \alpha \sin \beta, \\ \sin(\alpha + \beta) &= \cos \alpha \sin \beta + \sin \alpha \cos \beta. \\ \end{aligned}$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 $e^{i \phi} = \cos \phi + i \sin \phi$ and $i^2 = -1$. Then you get

$\begin{aligned} \cos(\alpha + \beta) + i \sin(\alpha + \beta) &= e^{i (\alpha+\beta)} \\ &= e^{i \alpha} e^{i \beta} \\ &= (\cos \alpha + i \sin \alpha)(\cos \beta + i \sin \beta) \\ &= (\cos \alpha \cos \beta - \sin \alpha \sin \beta) + i (\cos \alpha \sin \beta + \sin \alpha \cos \beta). \end{aligned}$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 $\alpha$-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?