91Ó°ÊÓ

The Pythagorean identity is a fundamental relation in trigonometry. At its core, it connects the squares of the sine, cosine, and tangent functions to the number 1. Specifically, the identity states that for any angle \( \theta \), \( \sin^2(\theta) + \cos^2(\theta) = 1 \). However, there are different forms of the Pythagorean identity, derived from the reciprocal identities of sine and cosine, which are secant \( (\sec) \) and cosecant \( (\csc) \) respectively.

One of these forms involves the secant and tangent functions, as \( \sec^2(\theta) = 1 + \tan^2(\theta) \), which becomes essential when simplifying trigonometric expressions as seen in the exercise. By rearranging this identity, we can also express \( \tan^2(\theta) \) in terms of \( \sec^2(\theta) \) by subtracting one from both sides, resulting in \( \tan^2(\theta) = \sec^2(\theta) - 1 \). Understanding and applying these identities requires practice, but they serve as powerful tools for solving and simplifying a wide range of trigonometric equations.

The secant function, denoted as \( \sec(\theta) \), is one of the six fundamental trigonometric functions. It's the reciprocal of the cosine function, meaning \( \sec(\theta) = \frac{1}{\cos(\theta)} \). Unlike the cosine function, which oscillates between -1 and 1, the secant function can take any real value, excluding the range between -1 and 1. This happens because secant tends to approach infinity (\