In trigonometry, simplify and find the exact value of (cos 7A + cos 5A) / (sin 7A - sin 5A).

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:

• Expression: (cos 7A + cos 5A) / (sin 7A - sin 5A)
• Use identities:
  • cos C + cos D = 2*cos((C + D)/2)*cos((C - D)/2)
  • sin C - sin D = 2*cos((C + D)/2)*sin((C - D)/2)

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:

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:

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