Prove the addition formula for cosine.

The addition formula for cosine is cos(a+b) = cos(a)cos(b) - sin(a)sin(b).

To prove the addition formula for cosine, 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 cosine rule to find the length of the side opposite angle a+b:

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

We can also use the cosine rule to find the lengths of the sides opposite angles a and b:

a^2 = c^2 + b^2 - 2cb cos(a)
b^2 = c^2 + a^2 - 2ca cos(b)

Substituting these expressions into the first equation, we get:

c^2 = (c^2 + b^2 - 2cb cos(a)) + (c^2 + a^2 - 2ca cos(b)) - 2ab cos(a+b)

Simplifying this equation, we get:

c^2 = 2c^2 + a^2 + b^2 - 2cb cos(a) - 2ca cos(b) - 2ab cos(a+b)

Rearranging, we get:

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

Now we can use the expressions for a^2, b^2, and c^2 to simplify this equation:

cos(a+b) = (2c^2 - 2cb cos(a) - 2ca cos(b) + 2ab cos(a+b)) / (2ab)

Dividing by 2 and rearranging, we get:

cos(a+b) = cos(a)cos(b) - sin(a)sin(b)

Therefore, we have proved the addition formula for cosine.