Simplify each expression. See Example 4.
tan 34°/2(1 - tan² 34°)

Verified step by step guidance

1. Recognize that the expression resembles the double-angle formula for tangent. Recall the formula: \(\tan(2\theta) = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).
2. Compare the given expression \(\frac{\tan 34^\circ}{2(1 - \tan^2 34^\circ)}\) with the double-angle formula. Notice that the numerator and denominator are arranged differently.
3. Rewrite the given expression to see if it can be manipulated into the form of the double-angle formula. For example, consider multiplying numerator and denominator appropriately or factoring to match the formula's structure.
4. Use the double-angle identity to express \(\tan 68^\circ\) in terms of \(\tan 34^\circ\) and relate it back to the given expression.
5. Conclude the simplification by substituting the equivalent double-angle expression, which will simplify the original expression into a single tangent function or a simpler trigonometric expression.

Key Concepts

Tangent Double-Angle Identity

The tangent double-angle identity expresses tan(2θ) in terms of tan(θ): tan(2θ) = 2 tan(θ) / (1 - tan²(θ)). This identity helps simplify expressions involving tangent of multiple angles by rewriting them in terms of a single angle.

Algebraic Simplification of Trigonometric Expressions

Simplifying trigonometric expressions often involves factoring, combining like terms, and applying identities. Recognizing patterns such as the double-angle formula allows rewriting complex fractions into simpler forms.

Evaluating Trigonometric Functions at Specific Angles

Understanding how to evaluate or manipulate trigonometric functions at given angles, like 34°, is essential. This includes knowing approximate values or using identities to rewrite expressions for easier calculation or simplification.