For an acute angle A, suppose (1 + cos A) / 2 = x. Using half angle trigonometric identities, determine which expression in terms of A is equal to x.

Introduction / Context:
This question tests your understanding of half angle identities in trigonometry. You are given that (1 + cos A) / 2 equals x for an acute angle A and asked which expression in terms of A also represents x. Knowing the standard half angle formulas that relate cosine of a double angle to squares of sine and cosine of half the angle is essential in simplifying trigonometric expressions and solving equations.

Given Data / Assumptions:

• A is an acute angle, so all basic trigonometric functions are positive.
• The expression (1 + cos A) / 2 is defined to be x.
• We must identify an equivalent expression in terms of A/2.
• Half angle identities for sine and cosine are available and valid.

Concept / Approach:
The relevant identity is the cosine half angle formula. For any angle θ, cos^2(θ) can be written in terms of cos(2θ) as cos^2(θ) = (1 + cos(2θ)) / 2. Here, if we let θ = A/2, then cos(2θ) becomes cos A. This gives a direct match between cos^2(A/2) and (1 + cos A) / 2. Recognising this pattern allows us to identify x as cos^2(A/2) without any complicated manipulation.

Step-by-Step Solution:
Step 1: Recall the identity for double angles: cos(2θ) = 2cos^2 θ − 1.Step 2: Rearrange this identity to express cos^2 θ: 2cos^2 θ = 1 + cos(2θ), so cos^2 θ = (1 + cos(2θ)) / 2.Step 3: Set θ = A/2. Then 2θ = A, and the identity becomes cos^2(A/2) = (1 + cos A) / 2.Step 4: Compare this with the given expression x = (1 + cos A) / 2.Step 5: It follows directly that x = cos^2(A/2).

Verification / Alternative check:
Consider a simple acute angle, for example A = 60°. Then cos 60° = 1/2, so (1 + cos 60°)/2 = (1 + 1/2)/2 = 3/2 ÷ 2 = 3/4. On the other hand, A/2 = 30°, and cos 30° = √3/2. Therefore cos^2(30°) = (√3/2)^2 = 3/4. The two values match, confirming that x equals cos^2(A/2) for this choice of A, and by identity it holds in general.

Why Other Options Are Wrong:
The expression sin(A/2) is not equal to (1 + cos A) / 2; instead, sin^2(A/2) is related to (1 − cos A) / 2. The square root expressions √(sin(A/2)) and √(cos(A/2)) involve fourth roots of trigonometric functions and do not match the given squared form. The expression sin^2(A/2) corresponds to (1 − cos A) / 2, not (1 + cos A) / 2. Only cos^2(A/2) exactly matches the formula for (1 + cos A) / 2.

Common Pitfalls:
Learners often mix up the half angle formulas, confusing the identities for sine and cosine. Another common error is to forget that the given expression is a square of a trigonometric function rather than the function itself. Careful use of the double angle identity and systematic substitution of θ = A/2 help to avoid these mistakes and to identify the correct equivalent expression.

Final Answer:
When (1 + cos A) / 2 = x, the equivalent expression is cos^2(A/2).