cos(a - B) + cos(a + B) = _____2sinasinB2cosacosB02(cosacosB + sinasinB)

The formula you're referring to is a trigonometric identity. The correct identity is:

cos(a - B) + cos(a + B) = 2cos(a)cos(B)

Here's the step by step derivation:

1. We start with the sum-to-product identities for cosine, which are:

cos(A - B) = cos(A)cos(B) + sin(A)sin(B) cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

2. We add these two equations together:

cos(A - B) + cos(A + B) = [cos(A)cos(B) + sin(A)sin(B)] + [cos(A)cos(B) - sin(A)sin(B)]

3. The sin(A)sin(B) terms cancel out, leaving us with:

cos(A - B) + cos(A + B) = 2cos(A)cos(B)

So, the correct answer to your question is 2cos(a)cos(B).