Use one or more of the six sum and difference identities to solve Exercises 13–54. In Exercises 13–24, find the exact value of each expression. tan ( 5𝝅/3 ﹣ 𝝅/4)

Verified step by step guidance

1. Identify the given expression: \(\tan\left( \frac{5\pi}{3} - \frac{\pi}{4} \right)\).
2. Recall the tangent difference identity: \(\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}\).
3. Set \(A = \frac{5\pi}{3}\) and \(B = \frac{\pi}{4}\), then find \(\tan A\) and \(\tan B\) separately.
4. Calculate \(\tan \frac{5\pi}{3}\) and \(\tan \frac{\pi}{4}\) using known values or reference angles.
5. Substitute these values into the tangent difference formula and simplify the resulting expression step-by-step.

Key Concepts

Sum and Difference Identities for Tangent

These identities express the tangent of a sum or difference of two angles in terms of the tangents of the individual angles. Specifically, tan(A - B) = (tan A - tan B) / (1 + tan A * tan B). This formula is essential for breaking down complex tangent expressions into simpler parts.

Exact Values of Trigonometric Functions at Special Angles

Certain angles like π/3, π/4, and π/5 have known exact trigonometric values involving square roots and rational numbers. Knowing these values allows for precise calculation without approximations, which is crucial when solving problems requiring exact answers.

Simplification of Trigonometric Expressions

After applying identities, simplifying the resulting expressions by combining like terms, rationalizing denominators, or reducing fractions is necessary. This step ensures the final answer is in its simplest exact form, which is often required in trigonometry exercises.