Ohio Assessments for Educators (OAE) Mathematics Practice Exam

Which of the following identities is derived from the Pythagorean Theorem?

1. tan²θ = sec²θ - 1

2. cot²θ + 1 = csc²θ

3. Both A and B

4. None of the above

The correct answer is: Both A and B

The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right triangle, typically expressed as \( a^2 + b^2 = c^2 \), where \( c \) is the hypotenuse. This theorem is the foundation for deriving various trigonometric identities. Both identities provided in the options can indeed be derived from the Pythagorean Theorem when applied in the context of right-angle triangles and the relationships between the sides and angles. The first identity, \( \tan^2θ = \sec^2θ - 1 \), stems from expressing tangent and secant in terms of sine and cosine. The identity can be understood through the Pythagorean relationship of the sides: since \( \secθ = \frac{1}{\cosθ} \), squaring it yields \( \sec^2θ = \frac{1}{\cos^2θ} \). By using the basic identity \( \sin^2θ + \cos^2θ = 1 \) rearranged as \( \tan^2θ + 1 = \sec^2θ \), we arrive at \( \tan^2θ = \sec^2