If √[(1 − cos A) / 2] = x for an angle A, identify which standard half angle trigonometric function of A is equal to x.

Introduction / Context:
This problem examines your understanding of half angle identities in trigonometry. These identities express trigonometric functions of half an angle in terms of functions of the full angle. Recognising the standard forms for sin(A/2) and cos(A/2) is important for simplifying expressions and solving equations, especially in exam questions that involve square roots and algebraic manipulation of trigonometric expressions.

Given Data / Assumptions:
- We are given that x = √[(1 − cos A) / 2].
- The angle A is such that the square root is defined and real, typically A between 0 and 360 degrees in standard contexts.
- We must decide which standard half angle function sin(A/2), cos(A/2), tan(A/2), or cot(A/2) matches this expression.

Concept / Approach:
The well known half angle formulas are:
sin(A/2) = ±√[(1 − cos A) / 2] and cos(A/2) = ±√[(1 + cos A) / 2].
The sign depends on the quadrant of A/2, but many exam questions assume an acute or principal value and focus only on the magnitude. Here, the presence of 1 − cos A inside the square root strongly suggests a connection with sin(A/2).

Step-by-Step Solution:
Step 1: Recall the identity sin(A/2) = ±√[(1 − cos A) / 2]. Step 2: Compare this with the given expression x = √[(1 − cos A) / 2]. Step 3: Notice that the algebraic form under the square root matches exactly: both have numerator 1 − cos A and denominator 2. Step 4: Because the question presents a principal square root and often in aptitude problems A is taken so that A/2 lies in a quadrant where sine is non negative, we take the positive branch of the identity. Step 5: Therefore x = sin(A/2) in the usual principal value sense. Step 6: Check the alternative half angle identity for cosine: cos(A/2) = ±√[(1 + cos A) / 2], which clearly has 1 + cos A instead of 1 − cos A, so it does not match.
Verification / Alternative check:
You can test the identity numerically with a specific angle. Take A = 60 degrees. Then cos 60° = 1/2. The expression becomes √[(1 − 1/2) / 2] = √[(1/2) / 2] = √(1/4) = 1/2. On the other hand, sin(A/2) = sin 30° = 1/2, which matches exactly. Cos(A/2) = cos 30° = √3 / 2 is different, confirming that the correct match is sin(A/2).

Why Other Options Are Wrong:
Option a, cos(A/2), uses the identity with 1 + cos A and therefore does not match the given expression.
Option b, tan(A/2), and option d, cot(A/2), have more complicated forms involving ratios of sine and cosine and cannot be written as a simple square root of (1 − cos A) / 2 without additional terms.
Option e, −sin(A/2), would only be correct in quadrants where sin(A/2) is negative, but the question uses the principal square root which is non negative, so x corresponds to sin(A/2), not its negative.

Common Pitfalls:
A frequent mistake is to confuse the signs in the half angle formulas or to swap the roles of sine and cosine, leading to misidentification of the expression. Some learners also forget that the plus or minus sign is determined by the quadrant of the half angle, but aptitude problems usually focus on the magnitude. Carefully memorising which formula has 1 − cos A and which has 1 + cos A is crucial to avoid these errors.

Final Answer:
The quantity x = √[(1 − cos A) / 2] is equal to sin(A/2).