Difficulty: Hard
Correct Answer: 1
Explanation:
Introduction / Context:This question tests trigonometric angle-shift identities (90°, 270°, 630°) and reciprocal functions (sec, cosec). These angles are designed so that everything reduces to simple sin A and cos A terms. After simplification, the expression becomes sin^2 A + cos^2 A, which is always 1.
Given Data / Assumptions:
Concept / Approach:Rewrite each trig function using standard shift identities, convert sec and cosec into cos and sin, then simplify each fraction separately and add.
Step-by-Step Solution:
Step 1: cos(90° + A) = -sin A. Step 2: cos(270° - A) = -sin A, so sec(270° - A) = 1/cos(270° - A) = 1/(-sin A) = -cosec A. Step 3: First term = (-sin A)/(-cosec A) = (-sin A)/(-1/sin A) = sin^2 A. Step 4: sin(270° + A) = -cos A. Step 5: sin(630° - A) = sin(270° - A) = -cos A, so cosec(630° - A) = 1/(-cos A) = -sec A. Step 6: Second term = (-cos A)/(-sec A) = (-cos A)/(-1/cos A) = cos^2 A. Step 7: Sum = sin^2 A + cos^2 A = 1.Verification / Alternative check:Take A = 0: first term becomes cos90/sec270 = 0/undefined? but through simplified identity form it becomes sin^2 0 = 0 and second term becomes cos^2 0 = 1, total 1 (valid where defined). The identity result remains 1 wherever the original expression is defined.
Why Other Options Are Wrong:
0 or -1: would contradict the identity sin^2 A + cos^2 A = 1. tan A*sec A or 3*sec A: involve sec, but the sec factors cancel during simplification.Common Pitfalls:Mis-evaluating 270° shifts (sign errors), and forgetting that sec and cosec are reciprocals, not independent functions.
Final Answer:1
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