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The secant function is one of the six fundamental trigonometric functions, just like sine and cosine. Its notation is \( \sec(x) \), and it is defined as the reciprocal of the cosine function. This means:

• \( \sec(x) = \frac{1}{\cos(x)} \)

Whenever we talk about the secant of an angle, we're essentially understanding what happens when the cosine gets very small or zero. In terms of right triangles, if cosine relates the adjacent side to the hypotenuse, then secant will show that relationship from the reverse perspective.
In trigonometric identities, the secant function often appears squared, such as \( \sec^2(x) \), because of its role in the Pythagorean identity:

• \( \sec^2(x) = 1 + \tan^2(x) \)

This identity is highly useful, allowing us to simplify and solve various trigonometric expressions, just like in the given problem. When simplifying expressions, recognizing how secant can be substituted helps to break down more complex expressions into simpler forms that can be compared or transformed easily.

The tangent function is another key player in the world of trigonometry. Notated as \( \tan(x) \), it is expressed as the ratio of the sine function to the cosine function:

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

Tangent primarily measures the steepness of a slope for a given angle \( x \) in a right triangle. Because of its unique properties, it continuously repeats its values, having periodicity over \( \pi \) (180 degrees).
In the context of trigonometric identities, the tangent function works well with other functions, such as the secant function. One of its most significant identities used in calculus and algebra is:

• \( 1 + \tan^2(x) = \sec^2(x) \)

Here, the tangent function lets us express much more complex angles in simpler terms, enabling efficient computation and verification of trigonometric expressions. In the original problem, \( \tan^2(x) \) and \( \tan^4(x) \) were manipulated to verify and simplify both sides of the given equation.

Simplifying trigonometric expressions is crucial to solving equations and verifying identities. In this process, we utilize known identities and algebraic manipulations to transform complex terms into simpler, more recognizable forms.

• **Step 1:** Identify the identity or definition applicable. Use identities like \( \sec^2(x) = 1 + \tan^2(x) \) to replace expressions appropriately.
• **Step 2:** Apply substitutions. Substitute each part of the expression systematically. Often, multiple substitutions may be needed to align terms properly.
• **Step 3:** Combine like terms and simplify. Group similar terms, and simplify them using algebraic operations to reach an easily interpretable form.

In our context, the exercise to verify the identity began with recognizing \( \sec^2(x) \) in both expressions and systematically substituting it with \( 1 + \tan^2(x) \). Furthermore, both sides were expanded and simplified step by step to end up with equal expressions: \( \tan^4 x + 2\tan^2 x + \tan^2 x \). By methodically following these steps, the expression was successfully verified as an identity.