Answer:
Explanation:
We can use the identity for the cosine of the difference of two angles, which states that:
cos(a - B) = cos(a)cos(B) + sin(a)sin(B)
If we rearrange this identity and subtract cos(B)cos(a) from both sides, we get:
cos(a - B) - cos(B)cos(a) = sin(a)sin(B)
Now, let's consider the expression cos(B) - cos(a). We can write this as:
cos(B) - cos(a) = - (cos(a) - cos(B))
Using the identity for the cosine of the sum of two angles, which states that:
cos(a + B) = cos(a)cos(B) - sin(a)sin(B)
We can rewrite the expression above as:
cos(B) - cos(a) = -2sin((a+B)/2)sin((a-B)/2)
Now, if we substitute (a - B) for a in the identity for the cosine of the difference of two angles that we derived earlier, we get:
cos(a - B) - cos(B)cos(a) = sin(a)sin(B)
And if we substitute (B - a) for B in the expression we derived for cos(B) - cos(a), we get:
cos(B) - cos(a) = -2sin((B+a)/2)sin((B-a)/2)
If we compare these two expressions, we see that they have the same sine term on the right-hand side. Therefore, we can conclude that:
cos(a - B) - cos(B)cos(a) = cos(B) - cos(a)
Or equivalently:
cos(a - B) = cos(B) - cos(a) + cos(B)cos(a)
This result shows that the values of cos(a - B) and cos(B) - cos(a) are related through the product of their cosine terms and a constant offset.