Simplify the trigonometric expression [sin A / (1 + cos A)] + [(1 + cos A) / sin A] and determine its final value in terms of a basic trigonometric function of A.

Introduction / Context:
This problem focuses on simplifying a compound trigonometric expression that mixes sine and cosine in fractional form. Such questions are designed to test your fluency with algebraic manipulation of trigonometric functions and your comfort with combining fractions and applying fundamental identities.

Given Data / Assumptions:

• The expression is E = sin A / (1 + cos A) + (1 + cos A) / sin A.
• Angle A is such that sin A and 1 + cos A are non zero, so the expression is defined.
• We must express the final simplified result in terms of a single standard trigonometric function.

Concept / Approach:
Main ideas:

• Use a common denominator to add the two fractions.
• Apply the Pythagorean identity sin^2 A + cos^2 A = 1.
• Look for factors that cancel after simplification.
• Recognise that cosec A is 1 / sin A.

Step-by-Step Solution:
Let E = sin A / (1 + cos A) + (1 + cos A) / sin A. The common denominator is sin A (1 + cos A). Write the numerator as sin A * sin A + (1 + cos A) * (1 + cos A). This gives sin^2 A + (1 + 2cos A + cos^2 A). Group terms: sin^2 A + cos^2 A + 1 + 2cos A. Use sin^2 A + cos^2 A = 1, so numerator becomes 1 + 1 + 2cos A = 2(1 + cos A). Thus, E = 2(1 + cos A) / [sin A (1 + cos A)]. Cancel the factor (1 + cos A) from numerator and denominator to get E = 2 / sin A = 2cosec A.

Verification / Alternative check:
Choose A = 30°. Then sin 30° = 1 / 2 and cos 30° = √3 / 2. Compute the original expression numerically and compare with 2cosec 30°. We get: sin A / (1 + cos A) = (1 / 2) / (1 + √3 / 2) and (1 + cos A) / sin A = (1 + √3 / 2) / (1 / 2). When evaluated carefully the total equals 4, and 2cosec 30° = 2 * 2 = 4, which matches.

Why Other Options Are Wrong:
Option a: 2sec A would equal 2 / cos A, which does not match the simplified form 2 / sin A.
Option c: 2tan A equals 2sin A / cos A and has a completely different dependence on A.
Option d: 2cot A is 2cos A / sin A, again not equal to 2 / sin A for general A.
Option e: 2 would require sin A = 1, which is not generally true for all A.

Common Pitfalls:
A frequent mistake is to incorrectly square (1 + cos A), or to forget to use the identity sin^2 A + cos^2 A = 1. Some students also cancel sin A or (1 + cos A) prematurely without having them as common factors in numerator and denominator. Keeping the algebra organized avoids these errors.

Final Answer:
The simplified value of the expression is 2cosec A.