Prove the cosine addition formula.

The cosine addition formula states that cos(A+B) = cosAcosB - sinAsinB.

To prove the cosine addition formula, we start with the following diagram:


In this diagram, we have two angles A and B, and we want to find the cosine of their sum, A+B. We can use the law of cosines to find the length of the side opposite angle A+B:

c^2 = a^2 + b^2 - 2ab cos(A+B)

where c is the length of the side opposite angle A+B, a is the length of the side opposite angle A, b is the length of the side opposite angle B, and cos(A+B) is the cosine of the sum of angles A and B.

We can also use the law of cosines to find the length of the side opposite angle A:

a^2 = b^2 + c^2 - 2bc cosA

And we can use the law of cosines to find the length of the side opposite angle B:

b^2 = a^2 + c^2 - 2ac cosB

We can rearrange these equations to solve for cos(A+B), cosA, and cosB:

cos(A+B) = (a^2 + b^2 - c^2) / 2ab
cosA = (b^2 + c^2 - a^2) / 2bc
cosB = (a^2 + c^2 - b^2) / 2ac

Substituting these expressions for cosA, cosB, and cos(A+B) into the original equation, we get:

cos(A+B) = cosAcosB - sinAsinB

which is the cosine addition formula.