Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which is the identity for cot(2θ)?

1. (cotθ-tanθ)/2

2. 1/tan(2θ)

3. (1-cos(2θ))/(sin(2θ))

4. tan²θ + 1

The correct answer is: (cotθ-tanθ)/2

The identity for cot(2θ) can be derived from the double angle formulas and the definition of cotangent. Cotangent is defined as the ratio of cosine to sine, so cot(2θ) can be expressed as: \[ \cot(2θ) = \frac{\cos(2θ)}{\sin(2θ)} \] Using the double angle formulas, we have: \[ \cos(2θ) = \cos^2(θ) - \sin^2(θ) \] and \[ \sin(2θ) = 2\sin(θ)\cos(θ) \] Now, substituting these into the expression for cot(2θ): \[ \cot(2θ) = \frac{\cos^2(θ) - \sin^2(θ)}{2\sin(θ)\cos(θ)} \] This expression can be simplified. Recognizing that \( \cot(θ) = \frac{\cos(θ)}{\sin(θ)} \) leads to the expression transforming in a way that illustrates: Using the identity for cotangent and tangent, we can rewrite cot(2θ) as follows. Knowing that: \[ \