Find one value of θ or x that satisfies each of the following.
cot(θ - 10°) = tan(2θ - 20°)

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Recall the identity that relates cotangent and tangent: \(\cot A = \tan(90^\circ - A)\). Use this to rewrite \(\cot(\theta - 10^\circ)\) as \(\tan\big(90^\circ - (\theta - 10^\circ)\big)\).

Rewrite the equation \(\cot(\theta - 10^\circ) = \tan(2\theta - 20^\circ)\) as \(\tan\big(90^\circ - (\theta - 10^\circ)\big) = \tan(2\theta - 20^\circ)\).

Simplify the expression inside the tangent on the left side: \(90^\circ - (\theta - 10^\circ) = 90^\circ - \theta + 10^\circ = 100^\circ - \theta\).

Set the arguments of the tangent functions equal to each other, considering the periodicity of tangent: \(100^\circ - \theta = 2\theta - 20^\circ + k \times 180^\circ\), where \(k\) is any integer.

Solve the resulting linear equation for \(\theta\) to find one or more values that satisfy the original equation.

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Relationship Between Cotangent and Tangent

Cotangent and tangent are reciprocal trigonometric functions, where cot(α) = 1/tan(α). Recognizing that cot(θ - 10°) can be rewritten as tan(90° - (θ - 10°)) helps transform the equation into a more solvable form by using complementary angle identities.

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Solving equations involving trigonometric functions often requires using identities, algebraic manipulation, and understanding periodicity. Equating angles or their related expressions and considering the periodic nature of tangent and cotangent functions is essential to find valid solutions.

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The identity tan(90° - x) = cot(x) links tangent and cotangent through complementary angles. Applying this identity allows rewriting cot(θ - 10°) as tan(100° - θ), enabling the equation to be expressed in terms of tangent functions for easier comparison and solution.

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