Simplify each expression. See Example 4.

2 tan 15°/(1 - tan² 15°)

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Recognize that the given expression matches the tangent double-angle identity, which is: \(\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).

Identify \(\theta\) in the expression. Here, \(\theta = 15^\circ\) because the expression is \(\frac{2 \tan 15^\circ}{1 - \tan^2 15^\circ}\).

Apply the double-angle formula by substituting \(\theta = 15^\circ\) into \(\tan(2\theta)\), so the expression simplifies to \(\tan(30^\circ)\).

Recall or use known values or properties of tangent to understand that \(\tan(30^\circ)\) is a standard angle value.

Conclude that the original expression simplifies to \(\tan(30^\circ)\), which can be further evaluated if needed.

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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.

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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.

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