Difficulty: Easy
Correct Answer: 2 cosec A
Explanation:
Introduction / Context:This item asks for algebraic simplification of a standard trigonometric sum involving sin A and cos A. The interval ensures all expressions are defined (excluding division by zero cases).
Given Data / Assumptions:
Concept / Approach:Put both terms over a common denominator, use 1 - cos^2 A = sin^2 A, and simplify carefully.
Step-by-Step Solution:
Common denominator = (1 + cos A)(1 - cos A) = 1 - cos^2 A = sin^2 A.Numerator = sin A (1 - cos A) + sin A (1 + cos A) = sin A (2) = 2 sin A.Thus value = (2 sin A) / (sin^2 A) = 2 / sin A = 2 cosec A.Verification / Alternative check:Pick A = 30°: LHS = (0.5/1.866...) + (0.5/0.134..) ≈ 0.268 + 3.732 ≈ 4.0; RHS = 2 * cosec 30° = 2 * 2 = 4 ✔️.
Why Other Options Are Wrong:2 sec A, 2 sin A, 2 cos A mismatch after the Pythagorean substitution and do not hold numerically for test angles.
Common Pitfalls:Forgetting the identity 1 - cos^2 A = sin^2 A; sign errors when expanding; canceling incorrectly before combining fractions.
Final Answer:2 cosec A
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