If tan(A/2) = x for an angle A, express x as a standard half angle identity in terms of sin A and cos A.

Introduction / Context:
This problem is designed to reinforce your knowledge of trigonometric half angle identities. The tan(A/2) formulas are especially important because they allow you to transform trigonometric expressions into rational functions of sine and cosine of the full angle. These identities are widely used in integration, equation solving, and simplification problems in competitive examinations.

Given Data / Assumptions:
- The half angle tangent is given as tan(A/2) = x.
- We must express x using only sin A and cos A, and match it with one of the given options.
- The angle A is assumed to be in a range where the expression is defined, typically avoiding values where denominators are zero.

Concept / Approach:
The standard half angle formulas for tangent are:
tan(A/2) = sin A / (1 + cos A) and tan(A/2) = (1 − cos A) / sin A.
Both are equivalent, and exam questions usually present one of these forms. We need to identify which option matches exactly one of these standard formulas, particularly the one using sin A in the numerator and 1 + cos A in the denominator.

Step-by-Step Solution:
Step 1: Recall the identity tan(A/2) = sin A / (1 + cos A). Step 2: Compare this with the answer choices. Option b is sin A / (1 + cos A), which matches the standard identity directly. Step 3: Another known form is tan(A/2) = (1 − cos A) / sin A, which is not listed in the options here, but confirms there are multiple equivalent ways to express tan(A/2). Step 4: Option a, sin A / (1 − cos A), resembles the reciprocal or a different combination and does not match the standard tan(A/2) identity. Step 5: Options c and d introduce square roots that are not part of the usual tan(A/2) formula, and option e is unrelated. Step 6: Therefore, the correct expression for x is sin A / (1 + cos A).
Verification / Alternative check:
You can verify the identity using the sine and cosine of half angles. Starting from the definitions, let sin(A/2) = s and cos(A/2) = c. Then tan(A/2) = s / c. Also, sin A = 2sc and cos A = c² − s². Hence sin A / (1 + cos A) becomes 2sc / (1 + c² − s²). Using c² + s² = 1, the denominator is 1 + c² − s² = 2c². This simplifies to (2sc) / (2c²) = s / c, which is tan(A/2). This derivation confirms the formula.

Why Other Options Are Wrong:
Option a, sin A / (1 − cos A), simplifies to a different expression that is more closely related to cot(A/2) after manipulation.
Options c and d add square roots around the ratio, which changes the value and does not match tan(A/2) directly.
Option e, cos A / (1 − sin A), has a different structure and is not a standard half angle identity for tangent.

Common Pitfalls:
A common mistake is confusing tan(A/2) with sin(A/2) or cos(A/2) or mixing up the signs in the numerator and denominator. Some learners also incorrectly place 1 − cos A in the denominator when the identity actually has 1 + cos A for this particular form. Remembering both forms, sin A / (1 + cos A) and (1 − cos A) / sin A, and knowing they are equivalent helps avoid confusion. Always check the structure carefully before choosing an answer.

Final Answer:
The correct standard identity is x = tan(A/2) = sin A / (1 + cos A).