Simplify each expression. See Example 4.


2 tan 15°/(1 - tan² 15°)

Verified step by step guidance

1. Recognize that the given expression matches the tangent double-angle identity, which is: \(\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).
2. Identify \(\theta\) in the expression. Here, \(\theta = 15^\circ\) because the expression is \(\frac{2 \tan 15^\circ}{1 - \tan^2 15^\circ}\).
3. Apply the double-angle formula by substituting \(\theta = 15^\circ\) into \(\tan(2\theta)\), so the expression simplifies to \(\tan(30^\circ)\).
4. Recall or use known values or properties of tangent to understand that \(\tan(30^\circ)\) is a standard angle value.
5. Conclude that the original expression simplifies to \(\tan(30^\circ)\), which can be further evaluated if needed.

Key Concepts

Tangent Double-Angle Identity

The tangent double-angle identity states that tan(2θ) = (2 tan θ) / (1 - tan² θ). This formula allows simplification of expressions involving tangent of double angles by relating them to the tangent of the original angle.

Simplification of Trigonometric Expressions

Simplifying trigonometric expressions involves applying identities and algebraic manipulation to rewrite expressions in simpler or more recognizable forms. This process often helps in solving equations or evaluating values more easily.

Evaluating Tangent at Specific Angles

Knowing or calculating the tangent of specific angles, such as 15°, is essential for evaluating expressions numerically or verifying simplified forms. This can be done using known angle values or by using sum and difference formulas.