Difficulty: Medium
Correct Answer: cot A
Explanation:
Introduction / Context:This question tests standard trigonometric sum-to-product identities. Expressions like cos C + cos D and sin C - sin D are designed so that the same common factor appears in numerator and denominator, allowing easy cancellation and producing a simple trig ratio (like tan or cot).
Given Data / Assumptions:
Concept / Approach:Convert both numerator and denominator into product form and cancel the common factor 2*cos((C + D)/2). Then simplify to a basic trig ratio.
Step-by-Step Solution:
Step 1: Numerator: cos 7A + cos 5A = 2*cos((7A + 5A)/2)*cos((7A - 5A)/2). Step 2: That becomes 2*cos(6A)*cos(A). Step 3: Denominator: sin 7A - sin 5A = 2*cos((7A + 5A)/2)*sin((7A - 5A)/2). Step 4: That becomes 2*cos(6A)*sin(A). Step 5: Divide: (2*cos(6A)*cos(A)) / (2*cos(6A)*sin(A)). Step 6: Cancel 2*cos(6A): result = cos(A)/sin(A) = cot A.Verification / Alternative check:Pick A = 10 degrees (any valid value): both sum-to-product forms remain valid and cancellation always leads to cot A, confirming the simplification is identity-based, not numeric coincidence.
Why Other Options Are Wrong:
tan A: inverse of the correct ratio. tan 4A or cot 4A: would appear if the half-sum or half-difference were 4A, but here they are 6A and A. sec A: not a quotient of sin and cos directly here.Common Pitfalls:Using the wrong identity for sin C - sin D (it uses cos of half-sum, not sin), or mixing up half-sum and half-difference.
Final Answer:cot A
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