Evaluate each expression. Give exact values. tan² 120° - 2 cot 240°

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Recall the definitions and values of the trigonometric functions involved: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \) and \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).

Find the exact value of \( \tan 120^\circ \). Use the fact that \( 120^\circ = 180^\circ - 60^\circ \), and recall the tangent function's behavior in the second quadrant.

Square the value of \( \tan 120^\circ \) to get \( \tan^2 120^\circ \).

Find the exact value of \( \cot 240^\circ \). Use the fact that \( 240^\circ = 180^\circ + 60^\circ \), and recall the cotangent function's behavior in the third quadrant.

Substitute the values found into the expression \( \tan^2 120^\circ - 2 \cot 240^\circ \) and simplify to get the exact value.

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Trigonometric Functions and Their Values

Understanding the basic trigonometric functions—sine, cosine, tangent, and cotangent—and their values at standard angles is essential. For angles like 120° and 240°, knowing how to find exact values using the unit circle or reference angles helps evaluate expressions precisely.

Reference Angles and Quadrants

Reference angles help determine the trigonometric function values for angles outside the first quadrant by relating them to acute angles. Knowing the sign of functions in each quadrant (e.g., tangent positive in Q3, negative in Q2) is crucial for correctly evaluating tan² 120° and cot 240°.

Trigonometric Identities and Simplification

Using identities such as cotangent being the reciprocal of tangent and the square of tangent (tan² θ) allows simplification of expressions. Recognizing these relationships helps combine and simplify terms like tan² 120° - 2 cot 240° to exact values.