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The cosine function is one of the fundamental elements of trigonometry, frequently used to describe the relationship between the sides of a right-angled triangle.
It is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. For any angle \( \theta \), the cosine function is represented as \( \text{cos}(\theta) \).

The graph of the cosine function shows a wave-like pattern which repeats every \(2\text{Ï€}\) radians (360 degrees), and it is symmetrical about the y-axis. This symmetry reflects the function's 'even' nature. Moreover, the cosine function has a range of values from -1 to 1. Understanding these properties can be vital for solving equations and proving identities involving trigonometry.

Trigonometric functions can be classified as even or odd, which is crucial when solving trigonometric equations or simplifying expressions.
The cosine function, as mentioned in the exercise, is an even function. This means that it satisfies the condition \( \text{cos}(-\theta) = \text{cos}(\theta) \). This property is derived from the function's symmetry about the y-axis. In contrast, functions like sine and tangent are odd functions, where \( \text{sin}(-\theta) = -\text{sin}(\theta) \) and \( \text{tan}(-\theta) = -\text{tan}(\theta) \).

When working with equations, knowing if a trigonometric function is even or odd helps simplify expressions and solve for unknown variables. For instance, in the textbook's step-by-step solution, knowing that the cosine function is even allowed for the simplification of \( \text{sec}(-t) \).

The secant function, represented as \( \text{sec}(θ) \), is the reciprocal of the cosine function, which makes it undefined whenever the cosine function equals zero.
It is defined as \( \text{sec}(θ) = \frac{1}{\text{cos}(θ)} \) and has a range that includes all real numbers except those that lie between -1 and 1, non-inclusive. Because \( \text{sec}(θ) \) is the reciprocal of an even function, it is also an even function, so \( \text{sec}(-θ) = \text{sec}(θ) \).

This property is vital to understanding how to handle the secant function in equations and identities. For example, if you know the value of \( \text{cos}(t) \), you can easily find \( \text{sec}(t) \) by taking its reciprocal, as demonstrated in the solution for calculating \( \text{sec}(-t) \). Understanding the secant function enhances a student's ability to work with complex trigonometric problems.