Final answer:
The given expression cos(255°) - cos(195°) is equal to 2. By using the sum-to-product formulas and solving for the angles involved, we determine the exact value of A is 1.
Step-by-step explanation:
To use the sum-to-product formulas we need to find two angles θ and φ such that θ+φ = 255° and θ-φ = 195°. By solving these equations, we can determine that θ = 225° and φ = 30°. The sum-to-product formula for cosine is:
cos(θ) - cos(φ) = -2sin((θ+φ)/2)sin((θ-φ)/2)
Substituting θ and φ into this formula, we get:
cos(225°) - cos(30°) = -2sin((225°+30°)/2)sin((225°-30°)/2)
cos(225°) - cos(30°) = -2sin(255°/2)sin(195°/2)
cos(225°) - cos(30°) = -2sin(127.5°)sin(97.5°)
Since cos(255°) - cos(195°) = 2 is given, it means:
-2sin(127.5°)sin(97.5°) = 2
By dividing both sides by -2, we find:
sin(127.5°)sin(97.5°) = -1
Therefore, the exact value of A is:
A = sin2(127.5°)sin2(97.5°)
A = (1)(1)
A = 1
The exact value of A is 1.