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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