Match each expression in Column I with its value in Column II.
(2 tan (π/3))/(1 - tan² (π/3))

Verified step by step guidance

1. Identify the trigonometric identity for the expression \( \frac{2 \tan(\theta)}{1 - \tan^2(\theta)} \), which is equivalent to \( \tan(2\theta) \).
2. Substitute \( \theta = \frac{\pi}{3} \) into the identity, so the expression becomes \( \tan(2 \times \frac{\pi}{3}) \).
3. Calculate \( 2 \times \frac{\pi}{3} \) to find the angle for which you need to determine the tangent.
4. Recognize that \( \tan(\frac{2\pi}{3}) \) is the tangent of an angle in the second quadrant.
5. Use the property that \( \tan(\pi - \theta) = -\tan(\theta) \) to find \( \tan(\frac{2\pi}{3}) \) by relating it to \( \tan(\frac{\pi}{3}) \).

Key Concepts

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(θ) = sin(θ)/cos(θ). Understanding the value of tan(π/3) is crucial for solving the given expression.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables. One important identity is the tangent double angle formula: tan(2θ) = (2 tan(θ))/(1 - tan²(θ)). This identity is directly applicable to the expression provided in the question, allowing for simplification and evaluation.

Radians and Degrees

Radians and degrees are two units for measuring angles. In trigonometry, angles are often expressed in radians, where π radians equals 180 degrees. The angle π/3 radians corresponds to 60 degrees, which is a key value for evaluating trigonometric functions like tangent. Recognizing this conversion is essential for accurately calculating the expression.