Find the exact value of each expression. See Example 1.
[tan 5π/12 + tan π/4]/[1 - tan 5π/12 tan π/4]

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Recognize that the given expression has the form of the tangent addition formula: \(\frac{\tan A + \tan B}{1 - \tan A \tan B} = \tan(A + B)\). Here, identify \(A = \frac{5\pi}{12}\) and \(B = \frac{\pi}{4}\).

Calculate the sum of the angles inside the tangent function: \(A + B = \frac{5\pi}{12} + \frac{\pi}{4}\). To add these, find a common denominator for the fractions.

Convert \(\frac{\pi}{4}\) to twelfths: \(\frac{\pi}{4} = \frac{3\pi}{12}\). Now add the angles: \(\frac{5\pi}{12} + \frac{3\pi}{12} = \frac{8\pi}{12}\).

Simplify the fraction \(\frac{8\pi}{12}\) to \(\frac{2\pi}{3}\). So, the expression simplifies to \(\tan\left(\frac{2\pi}{3}\right)\).

Recall or determine the exact value of \(\tan\left(\frac{2\pi}{3}\right)\) using the unit circle or reference angles to find the final exact value.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Tangent Addition Formula

The tangent addition formula states that tan(A + B) = (tan A + tan B) / (1 - tan A * tan B). This identity allows us to combine or break down tangent expressions involving sums of angles, which is essential for simplifying the given expression.

Exact Values of Special Angles

Certain angles like π/4 (45°) have well-known exact trigonometric values, such as tan(π/4) = 1. Recognizing these values helps in simplifying expressions without approximations, ensuring the answer is exact.

Angle Conversion and Decomposition

Angles like 5π/12 can be expressed as sums of special angles (e.g., π/3 + π/4). Decomposing complex angles into sums of known angles enables the use of addition formulas and exact values to find precise trigonometric values.

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