If √[(1 + cosA)/2] = x, then which trigonometric function of A correctly represents x?

Introduction / Context:
This trigonometry question focuses on half angle identities. We are given that x equals the square root of (1 + cosA)/2 and asked to identify which standard trigonometric function of A this corresponds to. Such identities are frequently used in simplifying expressions and solving trigonometric equations.

Given Data / Assumptions:

• √[(1 + cosA)/2] = x.
• A is an angle for which the standard half angle formulas apply.
• We must express x in terms of a basic trigonometric function of A/2.

Concept / Approach:
There are well known half angle formulas for sine and cosine. Specifically, cos(A/2) can be written as ±√[(1 + cosA)/2], while sin(A/2) can be written as ±√[(1 - cosA)/2]. For acute angles we usually take the positive square root. Therefore, the given expression √[(1 + cosA)/2] matches the formula for cos(A/2), and we can use this observation to pick the correct option directly.

Step-by-Step Solution:
Step 1: Recall the half angle formula for cosine: cos(A/2) = ±√[(1 + cosA)/2]. Step 2: Recall the half angle formula for sine: sin(A/2) = ±√[(1 - cosA)/2]. Step 3: Compare the given expression √[(1 + cosA)/2] with these standard forms. Step 4: We see that √[(1 + cosA)/2] matches cos(A/2) up to the sign choice. Step 5: For acute angles A, cos(A/2) is positive, so we take the positive root. Step 6: Therefore x = cos(A/2).

Verification / Alternative check:
Take A = 60 degrees. Then cos60 degrees = 1/2. The right side becomes √[(1 + 1/2)/2] = √[(3/2)/2] = √(3/4) = √3/2. Also, A/2 = 30 degrees and cos30 degrees = √3/2. Both evaluations give the same value, confirming the identity for this test angle.

Why Other Options Are Wrong:

• Option A: sin(A/2) corresponds to √[(1 - cosA)/2], not √[(1 + cosA)/2].
• Option B: tan(A/2) is expressed using (1 - cosA)/(1 + cosA) or sinA/(1 + cosA), not in this square root form.
• Option C: cot(A/2) involves the reciprocal of tan(A/2) and produces a different expression.
• Option E: sec(A/2) is the reciprocal of cos(A/2), so it is 1 / cos(A/2), not equal to cos(A/2).

Common Pitfalls:
A frequent mistake is to mix up the plus and minus signs that appear in half angle formulas, or to assign the formula for sin(A/2) to cos(A/2). Learners also sometimes forget that for acute angles the square roots are taken as positive, which simplifies the decision. Memorising the correct pair of identities and paying attention to the plus sign next to cosA helps greatly.

Final Answer:
Hence, the correct representation is x = cos(A/2).