Simplify the trigonometric expression (0° < A < 90°): Evaluate sin A / (1 + cos A) + sin A / (1 - cos A) and choose the correct simplified form.

Difficulty: Easy

2 cosec A
2 sec A
2 sin A
2 cos A

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:

Expression: sin A/(1 + cos A) + sin A/(1 - cos A).
Angle A in (0°, 90°).

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.

Simplify the trigonometric expression (0° < A < 90°): Evaluate sin A / (1 + cos A) + sin A / (1 - cos A) and choose the correct simplified form.

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