Which of the following is a Pythagorean identity?

The correct answer is found in the first choice, which states that sin²(θ) + cos²(θ) = 1. This is known as the Pythagorean identity because it is derived from the Pythagorean theorem, which relates the sides of a right triangle. In this identity, the sine and cosine functions represent the lengths of the legs of the triangle, while 1 corresponds to the square of the hypotenuse when the radius of the unit circle is 1. This foundational identity is crucial in trigonometry for simplifying expressions and solving equations involving sine and cosine.

The other choices describe important relationships in trigonometry, but they are not classified as Pythagorean identities. The second choice, sin(θ) + cos(θ) = 1, is a specific case that applies only under certain conditions, such as when θ is 45 degrees. The third choice, which states tan(θ) = sin(θ)/cos(θ), expresses the tangent function in terms of sine and cosine. While this is a valid trigonometric identity, it does not represent the sine and cosine relationship described by the Pythagorean identity. Therefore, the first option is the only correct representation of