Common Core State Standards

F-TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

It would be wonderfully convenient if the following two claims were true:

sin (x + y) = sin (x) + sin (y)

cos (x + y) = cos (x) + cos (y) .

As experimentation on a calculator shows (or just put in x = 30° and y = 60°) these dream formulas do not hold.

Can we find expressions for sin(x + y) and cos(x + y) nonetheless? You bet! We can use an approach similar to the one we used to prove the Pythagorean Theorem.

Draw two copies each of two right triangles, each with hypotenuse 1.

Arrange them into a rectangle as shown:

The area of the white space is the sum of the areas of two small rectangles:

White Space = sin(x) cos(y) + cos(x) sin(y).

We can also rearrange the four triangles within the large rectangle as

The white space is now a rhombus with side length 1. The area of a rhombus (in fact, of any parallelogram) is “base times height.” The base length is 1 and the height is the length h indicated. We see that h is the opposite edge of a right triangle of hypotenuse 1 and angle x + y. Thus:

White Space = 1 × h = 1 × sin(x + y) = sin(x + y).

It is the same white space. Thus we have proved:

sin(x + y) = sin x cos y + cos x sin y,

at least for angles x and y that lie between 0° and 90°.

(For visual ease, we've omitted displaying all the parentheses in this equation.)

In the same way, we can consider this variation of the proof. (Do you see the change on which angle is called y?)

It establishes

cos(x – y) = cos x cos y + sin x sin y,

at least for angles x and y that lie between 0° and 90°.