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To solve this problem, let's rewrite \(\sec(x)\) and \(\csc(x)\) in terms of \(\sin(x)\) and \(\cos(x)\), and then simplify.

### Step 1: Rewrite in terms of \(\sin(x)\) and \(\cos(x)\)
1. The secant function \(\sec(x)\) is defined as:
   \[
   \sec(x) = \frac{1}{\cos(x)}
   \]

2. The cosecant function \(\csc(x)\) is defined as:
   \[
   \csc(x) = \frac{1}{\sin(x)}
   \]

So, the expression \(\frac{\sec(x)}{\csc(x)}\) becomes:
\[
\frac{\sec(x)}{\csc(x)} = \frac{\frac{1}{\cos(x)}}{\frac{1}{\sin(x)}}
\]

### Step 2: Simplify the expression
To simplify, divide by \(\frac{1}{\sin(x)}\) by multiplying with its reciprocal:
\[
\frac{\frac{1}{\cos(x)}}{\frac{1}{\sin(x)}} = \frac{1}{\cos(x)} \cdot \frac{\sin(x)}{1} = \frac{\sin(x)}{\cos(x)}
\]

### Step 3: Recognize the simplified form
The result \(\frac{\sin(x)}{\cos(x)}\) is simply the tangent function:
\[
\frac{\sin(x)}{\cos(x)} = \tan(x)
\]

### Final Answer
\[
\frac{\sec(x)}{\csc(x)} = \tan(x)
\]

Would you like more details on any specific step or additional explanations?

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Here are some related questions you might find useful:
1. How do you derive the basic trigonometric identities for \(\sec(x)\) and \(\csc(x)\)?
2. What is the relationship between tangent and other trigonometric functions?
3. How do you simplify expressions involving \(\sec(x)\) and \(\csc(x)\) in other contexts?
4. Why does \(\tan(x) = \frac{\sin(x)}{\cos(x)}\)?
5. How would you simplify other combinations of trigonometric functions in terms of sine and cosine?

**Tip:** Memorizing the reciprocal and quotient identities of trigonometric functions can help simplify many problems more quickly.

Write the trigonometric expression \(\frac{\sec(x)}{\csc(x)}\) in terms of sine and cosine, and then simplify.

Learn how to simplify the trigonometric expression \(\frac{\sec(x)}{\csc(x)}\) by rewriting it in terms of sine and cosine. This solution covers key trigonometric identities and provides a step-by-step simplification, ideal for students in Grades 10-12.

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