91Ó°ÊÓ

The Pythagorean identity is fundamental in trigonometry. It states that for any angle \(\theta\), the following equation holds true: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \.\] This relationship is derived from the Pythagorean theorem, hence its name. By rearranging this identity, we can derive other useful forms: \[ \cos^2(\theta) = 1 - \sin^2(\theta) \text{ and } \sin^2(\theta) = 1 - \cos^2(\theta) \.\] These forms can help you solve various trigonometric equations and identities. Understanding the Pythagorean identity can simplify many trigonometric expressions.

The secant function, denoted as \(\text{sec}\), is the reciprocal of cosine. Mathematically, it is represented as: \[ \text{sec}(\theta) = \frac{1}{\text{cos}(\theta)}. \] Since \(\text{cos}(\theta)\) ranges between -1 and 1, \(\text{sec}(\theta)\) can take values less than -1 or greater than 1. The secant function is particularly useful in simplifying certain trigonometric identities. For instance, when you encounter \(\text{sec}^2(\theta) - 1\), you can use the Pythagorean identity to simplify this expression. One useful rearrangement from the Pythagorean identity is: \[ \text{sec}^2(\theta) = 1 + \text{tan}^2(\theta). \] This relationship often appears when dealing with secant and tangent being squared.

The tangent function, denoted as \tan\, is the ratio of the sine and cosine functions: \[ \tan(\theta) = \frac{\text{sin}(\theta)}{\text{cos}(\theta)}. \] The tangent function has many important properties and applications in trigonometry. It repeats every \(\theta = n \pi \) radians, where \(n\) is an integer, reflecting its periodic nature. One key relationship involving tangent is derived from the Pythagorean identity: \[ 1 + \text{tan}^2(\theta) = \text{sec}^2(\theta). \] This relationship can be used to simplify expressions or solve equations involving the tangent and secant functions. In the original problem, the expression \(\text{sec}^2(x) - 1\) simplifies to \(\text{tan}^2(x)\).

Cofunctions in trigonometry refer to pairs of trigonometric functions that are complementary to each other. For example, sine and cosine, tangent and cotangent are pairs of cofunctions. The basic identity for cofunctions is: \[ \text{sin}(90^\text{°} - \theta ) = \text{cos}(\theta)\. \] Similarly, \[ \text{cos}(90^\text{°} - \theta ) = \text{sin}(\theta)\. \] These identities are useful in solving problems involving right angles. By knowing the relationships between these cofunctions, you can transform or simplify trigonometric expressions. Another pair of cofunctions are secant and cosecant: \[ \text{sec}(90^\text{°} - \theta) = \text{csc}(\theta).\ \] Understanding these relationships helps in solving for angles or finding equivalent expressions in trigonometric identities.