Final answer:
To simplify the expression (cos A + cos B)² + (sin A + sin B)², trigonometric identities are used, leading to the confirmation that the expression equals 2[1 + cos(A - B)], thus the correct answer is 2.
Step-by-step explanation:
The question is asking to simplify the expression (cos A + cos B)² + (sin A + sin B)² and determine whether it equals 2[1 + cos(A - B)], with potential answers being 0, 1, 2, or 3. Using trigonometric identities, we can expand the squares and use the facts that sin² A + cos² A = 1 and 2 sin A cos B = sin(A + B) + sin(A - B), and similarly 2 cos A cos B = cos(A + B) + cos(A - B). Substituting these into the equation, we find that the given expression simplifies to 2[1 + cos(A - B)], which confirms that the statement is correct.
To simplify this, we get:
- (cos A + cos B)² = cos² A + 2 cos A cos B + cos² B
- (sin A + sin B)² = sin² A + 2 sin A sin B + sin² B
Adding these two expanded forms and using the Pythagorean identity yields:
- cos² A + cos² B + sin² A + sin² B + 2(cos A cos B + sin A sin B)
- 1 + 1 + 2 cos(A - B) which is equal to 2[1 + cos(A - B)]
The correct answer is option c) 2.