Simplify each expression. See Example 4.


2 tan 15°/(1 - tan² 15°)

Verified step by step guidance

1. Recognize that the expression \( \frac{2 \tan 15^\circ}{1 - \tan^2 15^\circ} \) resembles the double angle identity for tangent: \( \tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta} \).
2. Identify \( \theta = 15^\circ \) in the expression, which means the expression simplifies to \( \tan(2 \times 15^\circ) \).
3. Calculate \( 2 \times 15^\circ \) to find the angle for which you need to find the tangent.
4. Recognize that \( 2 \times 15^\circ = 30^\circ \).
5. Conclude that the expression simplifies to \( \tan 30^\circ \).

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 properties of the tangent function is essential for simplifying expressions involving angles.

Pythagorean Identity

The Pythagorean identity is a key relationship in trigonometry that states sin²(θ) + cos²(θ) = 1 for any angle θ. This identity can be manipulated to express tan²(θ) in terms of sine and cosine, specifically tan²(θ) = sin²(θ)/cos²(θ). This concept is crucial for simplifying expressions that involve squares of trigonometric functions.

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles in terms of single angles. For tangent, the formula is tan(2θ) = 2tan(θ)/(1 - tan²(θ)). This formula is particularly useful for simplifying expressions like the one given, as it allows for the transformation of the expression into a more manageable form.