Answer :
Let's go through the steps Melanie took and identify where the error occurred.
### Step 1
Melanie's initial expression is:
[tex]\[ \frac{\cot(x)}{\frac{1}{\sec(x)}} \][/tex]
First, let's rewrite cotangent and secant in terms of sine and cosine:
[tex]\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \quad \text{and} \quad \sec(x) = \frac{1}{\cos(x)} \][/tex]
Rewriting the expression:
[tex]\[ \frac{\cot(x)}{\frac{1}{\sec(x)}} = \frac{\frac{\cos(x)}{\sin(x)}}{\frac{1}{\frac{1}{\cos(x)}}} = \frac{\frac{\cos(x)}{\sin(x)}}{\cos(x)} \][/tex]
This matches Melanie's Step 1:
[tex]\[ \frac{\left(\frac{\cos(x)}{\sin(x)}\right)}{(\cos(x))} \][/tex]
So, Step 1 is correct.
### Step 2
Next, we simplify the fraction:
[tex]\[ \frac{\frac{\cos(x)}{\sin(x)}}{\cos(x)} = \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{\cos(x)} \][/tex]
This matches Melanie’s Step 2:
[tex]\[ \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{\cos(x)} \][/tex]
Step 2 is also correct.
### Step 3
Now, we can simplify the product:
[tex]\[ \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{\cos(x)} = \frac{1}{\sin(x)} \][/tex]
This matches Melanie's Step 3:
[tex]\[ \frac{1}{\sin(x)} \][/tex]
Step 3 is correct.
### Step 4
Finally, Melanie simplifies:
[tex]\[ \frac{1}{\sin(x)} \to \tan(x) \][/tex]
However, this step contains the mistake. The correct simplification of [tex]\(\frac{1}{\sin(x)}\)[/tex] is actually:
[tex]\[ \frac{1}{\sin(x)} = \csc(x) \][/tex]
not [tex]\(\tan(x)\)[/tex].
### Conclusion
Melanie made the first error in Step 4. The expression she should have written in that step is:
[tex]\[ \csc(x) \][/tex]
### Step 1
Melanie's initial expression is:
[tex]\[ \frac{\cot(x)}{\frac{1}{\sec(x)}} \][/tex]
First, let's rewrite cotangent and secant in terms of sine and cosine:
[tex]\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \quad \text{and} \quad \sec(x) = \frac{1}{\cos(x)} \][/tex]
Rewriting the expression:
[tex]\[ \frac{\cot(x)}{\frac{1}{\sec(x)}} = \frac{\frac{\cos(x)}{\sin(x)}}{\frac{1}{\frac{1}{\cos(x)}}} = \frac{\frac{\cos(x)}{\sin(x)}}{\cos(x)} \][/tex]
This matches Melanie's Step 1:
[tex]\[ \frac{\left(\frac{\cos(x)}{\sin(x)}\right)}{(\cos(x))} \][/tex]
So, Step 1 is correct.
### Step 2
Next, we simplify the fraction:
[tex]\[ \frac{\frac{\cos(x)}{\sin(x)}}{\cos(x)} = \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{\cos(x)} \][/tex]
This matches Melanie’s Step 2:
[tex]\[ \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{\cos(x)} \][/tex]
Step 2 is also correct.
### Step 3
Now, we can simplify the product:
[tex]\[ \frac{\cos(x)}{\sin(x)} \cdot \frac{1}{\cos(x)} = \frac{1}{\sin(x)} \][/tex]
This matches Melanie's Step 3:
[tex]\[ \frac{1}{\sin(x)} \][/tex]
Step 3 is correct.
### Step 4
Finally, Melanie simplifies:
[tex]\[ \frac{1}{\sin(x)} \to \tan(x) \][/tex]
However, this step contains the mistake. The correct simplification of [tex]\(\frac{1}{\sin(x)}\)[/tex] is actually:
[tex]\[ \frac{1}{\sin(x)} = \csc(x) \][/tex]
not [tex]\(\tan(x)\)[/tex].
### Conclusion
Melanie made the first error in Step 4. The expression she should have written in that step is:
[tex]\[ \csc(x) \][/tex]
