In a trigonometric identity problem, consider the expression [2 tan((π - A) / 2)] / [1 + tan^2((π - A) / 2)]. Using standard half angle identities, this expression can be simplified to a single basic trigonometric function of A. What is the simplified value?

Introduction / Context:
This question focuses on the use of trigonometric half angle identities. The given expression involves tan((π - A) / 2) inside a rational expression and asks you to simplify it to one of the basic trigonometric functions sin A, cos A or related forms. These types of simplifications are important in both pure trigonometry and in aptitude exams where one needs to quickly recognize patterns and identities.

Given Data / Assumptions:

• The expression to simplify is E = [2 tan((π - A) / 2)] / [1 + tan^2((π - A) / 2)].
• Angles are measured in radians, with π representing 180 degrees.
• A is such that all the trigonometric functions involved are defined.
• Standard trigonometric identities and cofunction properties apply.

Concept / Approach:
We recall two important ideas. First, there is a direct formula connecting sine of an angle with the tangent of its half: sin A = 2 tan(A / 2) / (1 + tan^2(A / 2)). Second, we use the cofunction identity for tangent: tan(π / 2 - θ) = cot θ. Noting that (π - A) / 2 = π / 2 - A / 2, we can relate tan((π - A) / 2) back to tan(A / 2) or to a similar expression and then match the pattern of the half angle identity for sine.

Step-by-Step Solution:
Step 1: Observe that (π - A) / 2 = π / 2 - A / 2. Step 2: Let T = tan((π - A) / 2). Then the expression becomes E = 2T / (1 + T^2). Step 3: Compare this with the standard identity sin θ = 2 tan(θ / 2) / (1 + tan^2(θ / 2)). Step 4: If we choose θ such that θ / 2 = (π - A) / 2, then θ = π - A. Step 5: Therefore, E = sin(π - A). Step 6: Use the supplementary angle identity sin(π - A) = sin A. Step 7: Hence, E simplifies to sin A.

Verification / Alternative check:
To verify, pick a simple numerical value, for example A = π / 3 (60 degrees). Compute the left side using a calculator: tan((π - π / 3) / 2) = tan(π / 3) and then evaluate 2 tan(π / 3) / (1 + tan^2(π / 3)). Independently compute sin(π / 3). Both computations give the same numerical value, confirming that the simplification to sin A is correct. Because the derivation is based on standard identities, the equality holds for all admissible values of A, not only for one specific choice.

Why Other Options Are Wrong:
Option a, 2 sin^2(A / 2), equals 1 - cos A and does not match the half angle tangent form used here. Option b, cos A, corresponds to (1 - tan^2(A / 2)) / (1 + tan^2(A / 2)), which is a different formula. Option d, 2 cos^2(A / 2), equals 1 + cos A and again does not fit the given structure. Option e, tan(A / 2), is only a half angle itself and not the result of the specific combination 2T / (1 + T^2). Only sin A matches the standard identity exactly after using the supplementary angle relation.

Common Pitfalls:
One common mistake is to misread the structure and think that the expression corresponds to a formula for cos A instead of sin A. Another frequent error is to ignore the presence of (π - A) / 2 and assume it is simply A / 2, which leads to confusion. It is important to recognize that sin(π - A) equals sin A, and this symmetry allows us to convert the expression to a neat form. Always rewrite the angle in a way that highlights known identities, and only then perform the simplification.

Final Answer:
Thus the simplified value of the given expression is sin A.