6. Use a graphing utility to graph the functions y1= sinx + cosx cot x and y2=csc x.
Make a conjecture about y1 and y2 Verify your result algbraially.


To tackle this problem, let's break it down into the following steps:

1. **Graph the Functions**: We will graph the functions $ y_1 = \sin(x) + \cos(x) \cot(x) $ and $ y_2 = \csc(x) $.
2. **Make a Conjecture**: Based on the graphs, we will form a conjecture about the relationship between $ y_1 $ and $ y_2 $.
3. **Verify Algebraically**: We will verify the conjecture using algebraic manipulation.

### Step 1: Graph the Functions

We need to graph:
\[ y_1 = \sin(x) + \cos(x) \cot(x) \]
\[ y_2 = \csc(x) \]

First, let's define $ \cot(x) $ and $ \csc(x) $:
\[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]
\[ \csc(x) = \frac{1}{\sin(x)} \]

Substituting $ \cot(x) $ into $ y_1 $, we get:
\[ y_1 = \sin(x) + \cos(x) \left( \frac{\cos(x)}{\sin(x)} \right) \]
\[ y_1 = \sin(x) + \frac{\cos^2(x)}{\sin(x)} \]

To graph these functions, let's proceed with the use of Python for plotting.

### Step 2: Make a Conjecture

Based on the graphs, we will make an educated guess about the relationship between $ y_1 $ and $ y_2 $.

### Step 3: Verify Algebraically

Using algebraic manipulation, we will verify if our conjecture holds true.

Let's start by plotting the graphs.

#### Graphing in Python:

```python
import numpy as np
import matplotlib.pyplot as plt

# Define the functions
def y1(x):
    return np.sin(x) + (np.cos(x)**2) / np.sin(x)

def y2(x):
    return 1 / np.sin(x)

# Create an array of x values
x = np.linspace(-2 * np.pi, 2 * np.pi, 1000)
# Avoid zero crossings for csc(x) to prevent division by zero
x = x[(x != 0) & (x != np.pi) & (x != -np.pi) & (x != 2 * np.pi) & (x != -2 * np.pi)]

# Compute y values
y1_values = y1(x)
y2_values = y2(x)

# Plot the functions
plt.figure(figsize=(10, 6))
plt.plot(x, y1_values, label='$y_1 = \sin(x) + \cos(x) \cot(x)$')
plt.plot(x, y2_values, label='$y_2 = \csc(x)$', linestyle='dashed')
plt.ylim(-10, 10)
plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5)
plt.legend()
plt.title('Graphs of $y_1$ and $y_2$')
plt.xlabel('x')
plt.ylabel('y')
plt.show()
```

### Step 2: Conjecture

From the graph, observe the behavior of $ y_1 $ and $ y_2 $. It appears that $ y_1 $ is equivalent to $ y_2 $. To conjecture, we propose:
\[ y_1 = y_2 \]

### Step 3: Verify Algebraically

Starting with $ y_1 $:
\[ y_1 = \sin(x) + \frac{\cos^2(x)}{\sin(x)} \]

Combine the terms:
\[ y_1 = \sin(x) + \frac{\cos^2(x)}{\sin(x)} = \frac{\sin^2(x)}{\sin(x)} + \frac{\cos^2(x)}{\sin(x)} = \frac{\sin^2(x) + \cos^2(x)}{\sin(x)} \]

Using the Pythagorean identity $ \sin^2(x) + \cos^2(x) = 1 $:
\[ y_1 = \frac{1}{\sin(x)} = \csc(x) \]

Thus, we have shown algebraically:
\[ y_1 = \csc(x) \]
Therefore, $ y_1 $ and $ y_2 $ are indeed equivalent.

Do you want details or have any questions? Here are 5 questions you might ask:

1. How do we graph functions with vertical asymptotes?
2. Can you explain the Pythagorean identity in more detail?
3. How does the domain affect the graph of $ y_1 $ and $ y_2 $?
4. What other trigonometric identities are useful for simplifying expressions?
5. Can we verify other trigonometric identities using a similar approach?

**Tip:** When dealing with trigonometric functions, always consider their domains and periodicity to understand their behavior better.

Learn how to graph trigonometric functions y1 = sin(x) + cos(x) cot(x) and y2 = csc(x) with detailed step-by-step instructions. Discover the relationship between these functions and verify algebraically using trigonometric identities. Suitable for students in Grades 10-12.

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