Difficulty: Medium
Correct Answer: 0
Explanation:
Introduction / Context:
This question checks your skill in simplifying expressions that involve powers of secant and tangent. The key is to use the fundamental Pythagorean identity sec^2 A = 1 + tan^2 A and to look for common factors and patterns. Such expressions often collapse to simple constants when manipulated correctly, a frequent trick in competitive examinations.
Given Data / Assumptions:
Concept / Approach:
The strategy is to expand and rearrange the expression so that sec^2 A and tan^2 A occur in factorable combinations. Group sec^4 A with sec^2 A and tan^4 A with tan^2 A. Recognising that sec^2 A − 1 equals tan^2 A, and that tan^2 A + 1 equals sec^2 A, helps you to simplify the grouped terms and see that they actually cancel each other out.
Step-by-Step Solution:
Verification / Alternative check:
To verify, choose a specific angle value where both sec A and tan A are defined, such as A = 45 degrees. Compute sec 45° = √2 and tan 45° = 1. Substitute into the original expression and evaluate numerically. You will find that the expression simplifies to zero, confirming the algebraic result.
Why Other Options Are Wrong:
The values −1, 1/2, 1, and 2 would require some nonzero residual expression after simplification. Since the final expression is sec^2 A * tan^2 A − sec^2 A * tan^2 A, any nonzero result indicates an algebraic mistake, such as dropping a sign or misapplying the identity sec^2 A = 1 + tan^2 A.
Common Pitfalls:
Common errors include incorrect distribution of the negative sign, misfactoring expressions like sec^4 A − sec^2 A, or writing sec^2 A = tan^2 A − 1 instead of tan^2 A + 1. Carefully grouping and factoring terms, and checking each identity used, avoids these traps and leads to the simple result.
Final Answer:
The expression (sec^4 A − tan^2 A) − (tan^4 A + sec^2 A) simplifies to 0.
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