Westonci.ca offers fast, accurate answers to your questions. Join our community and get the insights you need now. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
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]
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.