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We are tasked with simplifying the expression:

\[
\frac{5 \cos(x)}{(1 - \sin(x))(1 + \sin(x))}
\]

### Step 1: Simplify the denominator

Notice that the denominator is the product of two binomials: \( (1 - \sin(x))(1 + \sin(x)) \), which is a difference of squares. Using the identity \( (a - b)(a + b) = a^2 - b^2 \), we get:

\[
(1 - \sin(x))(1 + \sin(x)) = 1^2 - \sin^2(x) = 1 - \sin^2(x)
\]

### Step 2: Substitute the Pythagorean identity

We know from trigonometric identities that:

\[
\cos^2(x) = 1 - \sin^2(x)
\]

Therefore, the denominator becomes:

\[
1 - \sin^2(x) = \cos^2(x)
\]

### Step 3: Rewrite the entire expression

Now, substitute the simplified denominator back into the original expression:

\[
\frac{5 \cos(x)}{\cos^2(x)}
\]

### Step 4: Simplify the expression

Cancel out one factor of \( \cos(x) \) from both the numerator and the denominator:

\[
\frac{5}{\cos(x)} = 5 \sec(x)
\]

### Final Answer:
The simplified expression is \( 5 \sec(x) \), which corresponds to the first option.

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Would you like more details or have any other questions?

Here are 5 related questions:
1. What is the difference of squares identity and how is it used in simplifications?
2. How does the Pythagorean identity \( \sin^2(x) + \cos^2(x) = 1 \) help in simplifying trigonometric expressions?
3. What is the secant function, and how is it related to the cosine function?
4. How can you simplify other trigonometric fractions involving products of sine and cosine?
5. What are common trigonometric identities that are useful in simplifying expressions?

**Tip:** When simplifying trigonometric expressions, always look for identities such as the Pythagorean identities, as they often make the expression easier to handle.

Simplify the following expression: (5 cos(x)) / ((1 - sin(x))(1 + sin(x)))

Explore how to simplify the trigonometric expression 5 cos(x) / (1 - sin^2(x)) by using difference of squares and Pythagorean identities. This detailed guide is suitable for high school students in Grades 10-12, providing a step-by-step approach to understanding trigonometric identities and algebra.

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