Ohio Assessments for Educators (OAE) Mathematics Practice Exam

According to trigonometric identities, what is the relationship defined by the Pythagorean Theorem in trigonometry?

1. tan²θ + 1 = sec²θ

2. sin²θ + cos²θ = 1

3. 1 + cot²θ = csc²θ

4. All of the above

The correct answer is: All of the above

The Pythagorean Theorem in trigonometry is essential for understanding the relationships between the sine, cosine, tangent, and their corresponding secant, cosecant, and cotangent functions. The foundational identity, sin²θ + cos²θ = 1, directly derives from the Pythagorean Theorem, illustrating how the squares of the sine and cosine functions relate to the unity of a right triangle's sides. The first relationship, tan²θ + 1 = sec²θ, stems from dividing both sides of the sine and cosine identity by cos²θ. This provides another expression of the Pythagorean relationship involving tangent and secant. Similarly, the identity 1 + cot²θ = csc²θ illustrates a parallel relationship using cotangent and cosecant functions. Thus, all of these relationships are interconnected through the fundamental principles of trigonometry that hinge on the Pythagorean Theorem, supporting the conclusion that all the provided identities are valid expressions derived from this foundational theorem.