Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the formula for tan(2θ)?

1. 2tanθ/(1-tan²θ)

2. sin(2θ)/cos(2θ)

3. tan²θ + 1

4. 2sinθcosθ/(cos²θ - sin²θ)

The correct answer is: 2tanθ/(1-tan²θ)

The formula for tan(2θ) is derived from the double angle identities in trigonometry. Specifically, the tangent of a double angle can be expressed in terms of the tangent of the original angle. The formula states that: \[ \tan(2θ) = \frac{2\tan(θ)}{1 - \tan²(θ)} \] This identity is particularly useful because it allows one to find the tangent of double an angle directly using the value of the tangent of the original angle, making it easier to perform calculations without having to evaluate the sine and cosine functions separately. The other choices are related to trigonometric functions but do not correctly represent the tangent of the double angle. For instance, while sin(2θ) and cos(2θ) are related to the double angle identities, they are used to express sine and cosine, not tangent specifically. The expression for tan²θ + 1 refers to the Pythagorean identity relating tangent and secant, while the last option describes a different form that does not simplify to the standard tan(2θ) identity. Thus, the formula given correctly represents the tangent of double an angle through the tangent of the original angle.