Accuplacer Advanced Algebra and Functions Practice Exam

The correct relationship between the secant and tangent functions according to the Pythagorean identities is derived from the fundamental relationship between the sine and cosine functions. Specifically, the identity states that the square of the secant function, which is the reciprocal of the cosine function, is equal to the sum of the square of the tangent function, which is the ratio of the sine to cosine function, and one.

The identity can be expressed as:

\[ \sec^2(\theta) = 1 + \tan^2(\theta) \]

Rearranging this gives us:

\[ \sec^2(\theta) - \tan^2(\theta) = 1 \]

This shows that the secant squared minus the tangent squared equals one, effectively illustrating the relationship between these two functions.

The importance of this identity lies in its usefulness for simplifying expressions and solving equations that involve secant and tangent functions in trigonometry. Applications of this identity can often be found in calculus or physics problems involving angles and distances, where a deeper understanding of trigonometric relationships is required.

Each of the other options does not represent a valid Pythagorean identity involving secant and tangent, which makes them incorrect in this context.