Let's simplify the expression step by step:

$\frac{( \tan(x) + \sec(x) ) ( \tan(x) - \sec(x) )}{\sec(x)}$

Step 1: Use the difference of squares

Recall that the formula for the difference of squares is: $(a + b)(a - b) = a^2 - b^2$ In our case, we have $a = \tan(x)$ and $b = \sec(x)$. So, we can apply the formula: $(\tan(x) + \sec(x)) (\tan(x) - \sec(x)) = \tan^2(x) - \sec^2(x)$

Thus, the expression becomes: $\frac{\tan^2(x) - \sec^2(x)}{\sec(x)}$

Step 2: Use trigonometric identities

We know the following trigonometric identity: $\sec^2(x) = 1 + \tan^2(x)$ Substitute this into the numerator: $\tan^2(x) - \sec^2(x) = \tan^2(x) - (1 + \tan^2(x)) = -1$

So, the expression now simplifies to: $\frac{-1}{\sec(x)}$

Step 3: Simplify further

Recall that $\sec(x) = \frac{1}{\cos(x)}$, so: $\frac{-1}{\sec(x)} = -\cos(x)$

Final Answer:

$-\cos(x)$

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Do you need any further explanation on any of these steps, or have additional questions? Here are some related topics you might find useful:

1. How to simplify other trigonometric expressions.
2. The relationship between secant and cosine functions.
3. How the difference of squares formula is applied in trigonometry.
4. Understanding trigonometric identities in simplifying expressions.
5. What happens when we combine secant, tangent, and cosine in trigonometric identities?

Tip: When simplifying trigonometric expressions, always look for known identities like $\sec^2(x) = 1 + \tan^2(x)$ to help simplify complex expressions.