Find one value of θ or x that satisfies each of the following.
tan θ = cot(45° + 2θ)

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Recall the definition of cotangent in terms of tangent: \(\cot \alpha = \frac{1}{\tan \alpha}\). So the equation \(\tan \theta = \cot(45^\circ + 2\theta)\) can be rewritten as \(\tan \theta = \frac{1}{\tan(45^\circ + 2\theta)}\).

Multiply both sides of the equation by \(\tan(45^\circ + 2\theta)\) to get rid of the fraction: \(\tan \theta \cdot \tan(45^\circ + 2\theta) = 1\).

Use the tangent addition formula to express \(\tan(45^\circ + 2\theta)\): \(\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}\). Here, \(a = 45^\circ\) and \(b = 2\theta\), so \(\tan(45^\circ + 2\theta) = \frac{\tan 45^\circ + \tan 2\theta}{1 - \tan 45^\circ \tan 2\theta}\).

Substitute \(\tan 45^\circ = 1\) into the expression to simplify: \(\tan(45^\circ + 2\theta) = \frac{1 + \tan 2\theta}{1 - \tan 2\theta}\).

Replace \(\tan(45^\circ + 2\theta)\) in the equation \(\tan \theta \cdot \tan(45^\circ + 2\theta) = 1\) with the simplified expression and solve for \(\theta\): \(\tan \theta \cdot \frac{1 + \tan 2\theta}{1 - \tan 2\theta} = 1\). From here, you can proceed to isolate \(\theta\).

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