Match each expression in Column I with its value in Column II.
(2 tan 15°)/(1 - tan² 15°)

Verified step by step guidance

1. Recognize that the given expression \( \frac{2 \tan 15^\circ}{1 - \tan^2 15^\circ} \) matches the double-angle formula for tangent, which is \( \tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).
2. Identify \( \theta = 15^\circ \) in the expression, so the expression simplifies to \( \tan 2 \times 15^\circ = \tan 30^\circ \).
3. Recall the exact value of \( \tan 30^\circ \), which is a well-known special angle in trigonometry.
4. Use this value to match the expression from Column I with the corresponding value in Column II.
5. Confirm the match by verifying the angle and the formula used, ensuring the correct pairing.

Key Concepts

Tangent Double-Angle Formula

The tangent double-angle formula states that tan(2θ) = (2 tan θ) / (1 - tan² θ). This identity allows us to express the tangent of twice an angle in terms of the tangent of the original angle, simplifying calculations involving angle multiples.

Properties of the Tangent Function

The tangent function relates an angle in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic with period 180°, and understanding its behavior and values at special angles like 15° is essential for evaluating expressions.

Angle Measurement and Conversion

Angles can be measured in degrees or radians, and recognizing common angle values such as 15°, 30°, and 45° helps in applying trigonometric identities. Accurate angle measurement is crucial for correctly using formulas and matching expressions to their values.