Use the given information to find the exact value of each of the following: tan 2θ
cot θ = 2, θ lies in quadrant III.

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Recall the double-angle identity for tangent: \(\tan 2\theta = \frac{2 \tan \theta}{1 - \tan^2 \theta}\).

Substitute \(\tan 2\theta\) into the expression \(\tan 2\theta \cot \theta\) to get \(\left( \frac{2 \tan \theta}{1 - \tan^2 \theta} \right) \cdot \cot \theta\).

Rewrite \(\cot \theta\) as \(\frac{1}{\tan \theta}\) and simplify the expression: \(\left( \frac{2 \tan \theta}{1 - \tan^2 \theta} \right) \cdot \frac{1}{\tan \theta} = \frac{2}{1 - \tan^2 \theta}\).

Set the simplified expression equal to 2 (given): \(\frac{2}{1 - \tan^2 \theta} = 2\) and solve for \(\tan^2 \theta\).

Use the fact that \(\theta\) lies in quadrant III, where tangent is positive, to determine the exact value of \(\tan \theta\) from \(\tan^2 \theta\).

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Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. For example, the double-angle identity for tangent, tan(2θ) = 2 tan θ / (1 - tan² θ), is essential for expressing tan 2θ in terms of tan θ, enabling simplification and solving of equations.

Quadrant Sign Rules

The sign of trigonometric functions depends on the quadrant in which the angle lies. In quadrant III, both sine and cosine are negative, making tangent positive. Understanding these sign conventions is crucial for determining the correct values of trigonometric functions and their ratios.

Relationship Between Tangent and Cotangent

Tangent and cotangent are reciprocal functions: cot θ = 1 / tan θ. This relationship allows conversion between the two, simplifying expressions like tan 2θ cot θ. Recognizing this reciprocal nature helps in manipulating and solving trigonometric equations efficiently.

Use the given information to find the exact value of each of the following: tan 2θ cot θ = 2, θ lies in quadrant III.