cos s + t = cos s cos t - sin s sin t

sin s + t = sin s cos t + cos s sin t

For example, here is one using Euler's formula, and here is a derivation of the difference version of the cosine that I encountered in Gil Strang's Calculus textbook. As I've said, I find it only necessary to memorize one:

cos s - t = cos s cos t + sin s sin t

which is easily checked for the case where s = t.

Recently I ran into a couple more. Probably the simplest is this one:

Two stacked right triangles with a surrounding rectangle. The upper triangle is sized so that the hypotenuse is 1 and the sine and cosine are obvious. For the triangle with angle phi, we need an extra term in the sine and cosine so that when dividing, say, opposite/hypotenuse, the result is correct.

The angle theta + phi is known by the alternate interior angles theorem, and the small triangle with angle phi is known by a combination of the complementary and supplementary angles theorems.

Now, just read the result. One diagram gives both formulas!

There is also a simple derivation for the sine formula based on area calculations. We calculate the area of this triangle in two ways:

On the left, we have that

A = (1/2) a sin (theta + phi) b

On the right (h = a cos phi = b cos theta) and

A = (1/2) h a sin phi + 1/2 h b sin theta

= (1/2) (b cos theta a sin phi + a cos phi b sin theta)

Equating the two and factoring out (1/2)ab, we obtain the addition formula for sine. (Unfortunately, I've lost the links where I saw these).