If `"cot" theta "cot" 7 theta + "cot" theta "cot" 4 theta + "cot" 4 theta "cot" 7 theta = 1, " then" theta =`

Options

A. `n pi, n in Z`
B. `(2n +1) (pi)/(2), n in Z`
C. `n pi + (-1)^(n) (pi)/(2), n inZ`
D. `(n pi)/(12), n in Z`

Answer

To solve the equation \( \cot \theta \cot 7\theta + \cot \theta \cot 4\theta + \cot 4\theta \cot 7\theta = 1 \), we will follow these steps:

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \cot \theta \cot 7\theta + \cot \theta \cot 4\theta + \cot 4\theta \cot 7\theta = 1 \]
We can rearrange it to isolate one of the terms. Let's factor out \( \cot 7\theta \):
\[ \cot 7\theta (\cot \theta + \cot 4\theta) = 1 - \cot \theta \cot 4\theta \]

### Step 2: Expressing \( \cot 7\theta \)
From the rearranged equation, we can express \( \cot 7\theta \) as:
\[ \cot 7\theta = \frac{1 - \cot \theta \cot 4\theta}{\cot \theta + \cot 4\theta} \]

### Step 3: Using the Cotangent Addition Formula
Using the cotangent addition formula:
\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]
we can apply this to our expression. Here, let \( a = \theta \) and \( b = 4\theta \):
\[ \cot(5\theta) = \frac{\cot \theta \cot 4\theta - 1}{\cot \theta + \cot 4\theta} \]

### Step 4: Setting the Equation
Now we can set the equation:
\[ \cot 7\theta = \cot(5\theta) \]
This implies:
\[ 7\theta = 5\theta + n\pi \quad \text{for some integer } n \]

### Step 5: Solving for \( \theta \)
Rearranging gives:
\[ 2\theta = n\pi \quad \Rightarrow \quad \theta = \frac{n\pi}{2} \]

### Step 6: Finding Specific Values
To find specific values of \( \theta \), we can substitute different integer values for \( n \):
- For \( n = 0 \): \( \theta = 0 \)
- For \( n = 1 \): \( \theta = \frac{\pi}{2} \)
- For \( n = 2 \): \( \theta = \pi \)
- For \( n = 3 \): \( \theta = \frac{3\pi}{2} \)
- And so on...

### Final Answer
Thus, the general solution for \( \theta \) is:
\[ \theta = \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]