If tan x = 1 for an acute angle x with 0° < x < 90°, then using standard trigonometric identities, what is the value of the expression 2 sin x cos x?

Introduction / Context:
This question checks your familiarity with standard trigonometric values and identities. When you know tan x for an acute angle, you should be able to infer sin x and cos x or use angle knowledge to evaluate expressions like 2 sin x cos x. The expression 2 sin x cos x is especially important because it is equal to sin 2x, a well known double angle identity.

Given Data / Assumptions:

• tan x = 1.
• 0° < x < 90°, so x is an acute angle.
• We need the value of 2 sin x cos x.

Concept / Approach:
First, recall which acute angle has tangent equal to 1. The standard angle with tan x = 1 is x = 45°. Once we know x, we can either directly use the identity 2 sin x cos x = sin 2x, or substitute the well known values sin 45° and cos 45°. Using the double angle identity is faster because it turns the problem into evaluating sin of a specific known angle.

Step-by-Step Solution:
Given tan x = 1 and x is acute, we know x = 45°. Recall the double angle identity: sin 2x = 2 sin x cos x. We need 2 sin x cos x, which is exactly sin 2x. Compute 2x when x = 45°: 2x = 2 * 45° = 90°. Therefore, 2 sin x cos x = sin 2x = sin 90°. The value of sin 90° is 1. Hence, 2 sin x cos x = 1.

Verification / Alternative check:
We can also compute sin x and cos x explicitly. For x = 45°, sin 45° = sqrt(2) / 2 and cos 45° = sqrt(2) / 2. Then 2 sin x cos x = 2 * (sqrt(2) / 2) * (sqrt(2) / 2) = 2 * (2 / 4) = 2 * (1 / 2) = 1. This confirms the result obtained via the double angle identity.

Why Other Options Are Wrong:
1/2 and sqrt(3) / 2 are typical sine values of 30° and 60°, not 90°. sqrt(3) and 0 do not correspond to sin 2x for any acute x in this setting. Since the expression exactly equals sin 90°, and sin 90° is 1, none of the other options fit.

Common Pitfalls:
Some students confuse the angle whose tangent is 1 and mistakenly use 30° or 60°. Others forget the double angle identity and attempt to compute sin x and cos x from tan x using a right triangle, which is valid but longer. Remembering that 2 sin x cos x = sin 2x is a powerful shortcut in many trigonometry problems.

Final Answer:
The value of 2 sin x cos x is 1.