Difficulty: Medium
Correct Answer: (1 - tan A)^2 + (1 + tan A)^2
Explanation:
Introduction / Context: This question tests transforming expressions using basic trigonometric identities and algebraic expansion. A common identity is sec^2 A = 1 + tan^2 A. If we multiply both sides by 2, we get 2 sec^2 A = 2(1 + tan^2 A). The answer options are written in expanded-square forms, so the task is to expand and simplify those forms and see which one matches 2(1 + tan^2 A) exactly. This checks both trig knowledge and careful algebra.
Given Data / Assumptions:
Concept / Approach: Compute 2 sec^2 A using the identity: 2 sec^2 A = 2(1 + tan^2 A). Then expand candidate expressions like (1 - tan A)^2 and (1 + tan A)^2: (1 - t)^2 = 1 - 2t + t^2 (1 + t)^2 = 1 + 2t + t^2 Adding them cancels the middle terms and produces 2 + 2t^2 = 2(1 + t^2), which matches 2 sec^2 A when t = tan A.
Step-by-Step Solution: 1) Start with identity: sec^2 A = 1 + tan^2 A 2) Multiply by 2: 2 sec^2 A = 2(1 + tan^2 A) 3) Let t = tan A for easier algebra 4) Expand (1 - t)^2: (1 - t)^2 = 1 - 2t + t^2 5) Expand (1 + t)^2: (1 + t)^2 = 1 + 2t + t^2 6) Add the two expansions: (1 - t)^2 + (1 + t)^2 = (1 - 2t + t^2) + (1 + 2t + t^2) 7) Simplify: = 2 + 2t^2 = 2(1 + t^2) 8) Replace t with tan A: = 2(1 + tan^2 A) = 2 sec^2 A
Verification / Alternative check: Take A = 45°: tan A = 1, sec A = √2, so 2 sec^2 A = 2*(2) = 4. Option (1 - tan A)^2 + (1 + tan A)^2 becomes (1 - 1)^2 + (1 + 1)^2 = 0 + 4 = 4. Matches exactly.
Why Other Options Are Wrong: • (1 - tan A)^2 - (1 + tan A)^2 simplifies to -4tan A, not 2sec^2 A. • Square-root options change the expression’s value and do not equal 2(1 + tan^2 A). • (1 + tan A)^2 alone expands to 1 + 2tan A + tan^2 A, which does not match 2 + 2tan^2 A.
Common Pitfalls: • Forgetting sec^2 A = 1 + tan^2 A and using sec A = 1/cos A without simplifying. • Expanding (1 ± tan A)^2 incorrectly or missing the cancellation when adding.
Final Answer: (1 - tan A)^2 + (1 + tan A)^2
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