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 ( 𝝅/3 + 𝝅/4 )

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Identify the given expression: \(\tan\left( \frac{\pi}{3} + \frac{\pi}{4} \right)\).

Recall the tangent sum identity: \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\).

Set \(A = \frac{\pi}{3}\) and \(B = \frac{\pi}{4}\), then substitute into the identity: \(\tan\left( \frac{\pi}{3} + \frac{\pi}{4} \right) = \frac{\tan \frac{\pi}{3} + \tan \frac{\pi}{4}}{1 - \tan \frac{\pi}{3} \tan \frac{\pi}{4}}\).

Find the exact values of \(\tan \frac{\pi}{3}\) and \(\tan \frac{\pi}{4}\) using known special angles: \(\tan \frac{\pi}{3} = \sqrt{3}\) and \(\tan \frac{\pi}{4} = 1\).

Substitute these values back into the formula and simplify the numerator and denominator separately to express the exact value of the tangent.

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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). They are essential for breaking down complex angle expressions into simpler parts to find exact values.

Exact Values of Trigonometric Functions at Special Angles

Certain angles like π/3 and π/4 have well-known exact trigonometric values (e.g., tan(π/3) = √3, tan(π/4) = 1). Knowing these values allows for precise calculation without approximations, which is crucial when applying sum and difference identities to find exact results.

Simplification of Rational Expressions

After applying the sum or difference identity, the resulting expression often involves fractions and radicals. Simplifying these rational expressions correctly is necessary to arrive at the exact value in its simplest form, ensuring clarity and correctness in the final answer.