If sec^2 A * cosec^2 A = x, which expression is equal to x?

Introduction / Context:This question tests rewriting sec^2 and cosec^2 in terms of tan^2 and cot^2 using Pythagorean identities. Because sec^2 A = 1 + tan^2 A and cosec^2 A = 1 + cot^2 A, their product expands neatly, and the hidden key step is noticing that tan^2 A * cot^2 A = 1 (where defined).

Given Data / Assumptions:

• sec^2 A * cosec^2 A = x
• Identities:
  • sec^2 A = 1 + tan^2 A
  • cosec^2 A = 1 + cot^2 A
  • tan A = sin A / cos A and cot A = cos A / sin A, so tan A * cot A = 1

Concept / Approach:Substitute sec^2 A and cosec^2 A in terms of tan^2 A and cot^2 A, expand the product, and simplify using tan^2 A * cot^2 A = 1.

Step-by-Step Solution:

Verification / Alternative check:Let A = 45°: sec^2 45° = 2 and cosec^2 45° = 2, product x = 4. RHS: tan^2 45° + cot^2 45° + 2 = 1 + 1 + 2 = 4. Matches.

Why Other Options Are Wrong:

Common Pitfalls:Forgetting tan^2 A*cot^2 A = 1, or treating sec^2 as 1 - tan^2 (wrong sign).

Final Answer:tan^2 A + cot^2 A + 2