Find the exact value of each expression. See Example 1.
[tan 80° - tan(-55°)]/[ 1 + tan 80° tan(-55°)]

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Recognize that the given expression has the form of the tangent subtraction formula: \(\frac{\tan A - \tan B}{1 + \tan A \tan B} = \tan(A - B)\).

Identify the angles in the expression: \(A = 80^\circ\) and \(B = -55^\circ\).

Apply the tangent subtraction formula to rewrite the expression as \(\tan(80^\circ - (-55^\circ))\).

Simplify the angle inside the tangent function: \(80^\circ - (-55^\circ) = 80^\circ + 55^\circ\).

Express the final step as \(\tan(135^\circ)\), which is the exact value of the original expression.

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Key Concepts

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

Tangent Function and Its Properties

The tangent function, tan(θ), relates the angle θ in a right triangle to the ratio of the opposite side over the adjacent side. It is periodic with period 180° and odd, meaning tan(-θ) = -tan(θ). Understanding these properties helps simplify expressions involving negative angles.

Tangent Addition Formula

The tangent addition formula states that tan(A + B) = (tan A + tan B) / (1 - tan A tan B). This formula is essential for combining or simplifying expressions involving sums or differences of angles in tangent form.

Recognizing the Expression as a Tangent Difference

The given expression matches the form (tan A - tan B) / (1 + tan A tan B), which corresponds to tan(A - B). Recognizing this allows direct simplification by converting the expression into a single tangent of a difference of angles.

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