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The tangent function, denoted as \( \tan x \), is one of the basic trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle. Mathematically, this is expressed as \( \tan x = \frac{\sin x}{\cos x} \). Here, \( \sin x \) and \( \cos x \) represent the sine and cosine functions, respectively.
There are some key points that highlight the behavior of the tangent function:

• The tangent function is periodic with a period of \( \pi \). This means that every \( \pi \) radians, the function repeats its values.
• \( \tan x \) can take any real value as it ranges from \(-\infty\) to \(+\infty\).
• Tangent is undefined whenever \( \cos x = 0 \), which occurs at \( \frac{\pi}{2} + n\pi \) for any integer \( n \).

The tangent function is extremely useful in trigonometry. It simplifies expressions by relating sine and cosine. Understanding its properties helps in resolving trigonometric equations, as we see in our exercise when \( \tan x \) was replaced by its definition to simplify the expression.

The secant function, \( \sec x \), is another vital trigonometric function, closely related to the cosine function. It is the reciprocal of the cosine function and is defined as \( \sec x = \frac{1}{\cos x} \). Because it relies on the cosine function, understanding secant requires knowledge of cosine.
Some important characteristics of the secant function include:

• Like the tangent, the secant function is also periodic, with the same period of \( 2\pi \).
• It is undefined wherever \( \cos x = 0 \), leading to vertical asymptotes in its graph at these points.
• The secant function's range includes all real numbers greater than or equal to 1 or less than or equal to -1.

In our given exercise, we see how the secant function, as an even function, simplifies things when \( \sec(-x) \) equals \( \sec x \). This even property allows us to handle negative angles easily during trigonometric simplifications.

Understanding even and odd functions is crucial for working with trigonometric functions and expressions. An even function satisfies the condition \( f(-x) = f(x) \) for all \( x \) in its domain. Examples include \( \cos x \) and \( \sec x \). These functions exhibit symmetry about the y-axis.
On the other hand, an odd function satisfies \( f(-x) = -f(x) \) for all \( x \) in its domain. \( \sin x \) and \( \tan x \) are examples of odd functions, showing symmetry about the origin.

• Even functions do not change with the sign of the angle, simplifying expressions when negative angles are involved.
• Odd functions change sign with the angle, which affects how they contribute to expressions.

These properties proved pivotal when simplifying \( \frac{\tan x}{\sec(-x)} \). Recognizing \( \sec(-x) \) as an even function allowed for direct substitution without complicating the expression further. Understanding these concepts helps in accurately manipulating and simplifying diverse trigonometric expressions.