Introduction:
In this problem, we need to evaluate the expression 2tan(30°) / (1tan^2(30°)). To solve this, we will break down the expression into smaller parts and simplify each part individually.

Key Concepts:
Before we proceed, let's understand some key concepts related to trigonometric functions that will be useful in solving this problem:

1. Tangent Function (tan): The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side.

2. Trigonometric Identities: Trigonometric identities are equations that relate the trigonometric functions to each other. These identities help us simplify expressions involving trigonometric functions.

Solution:
Let's solve the given expression step by step:

Step 1: Evaluate tan(30°):
The tangent of 30 degrees can be calculated by considering a right triangle with one angle measuring 30 degrees. In this triangle, the side opposite to the 30-degree angle is 1, and the adjacent side is √3 (as per the properties of a 30-60-90 triangle).

So, tan(30°) = Opposite side / Adjacent side = 1 / √3 = √3 / 3.

Step 2: Evaluate tan^2(30°):
To find the square of tan(30°), we can simply square the value we calculated in step 1:

tan^2(30°) = (√3 / 3)^2 = 3 / 9 = 1 / 3.

Step 3: Substitute the values into the expression:
Now, let's substitute the values we calculated in steps 1 and 2 into the given expression:

2tan(30°) / (1tan^2(30°)) = 2(√3 / 3) / (1(1 / 3)) = 2√3 / (1/3) = 2√3 * 3/1 = 6√3.

Final Answer:
Therefore, the simplified form of the expression 2tan(30°) / (1tan^2(30°)) is 6√3.

Summary:
- The expression 2tan(30°) / (1tan^2(30°)) simplifies to 6√3.
- To solve this, we evaluated the tangent of 30 degrees (tan(30°)) and its square (tan^2(30°)).
- By substituting the values into the expression, we obtained the simplified form.

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