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 this equation by isolating one of the terms. Let's isolate \( \cot 7\theta \):
\[ \cot 7\theta (\cot \theta + \cot 4\theta) = 1 - \cot \theta \cot 4\theta \]

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

### Step 3: Using the Cotangent Addition Formula
Recall the cotangent addition formula:
\[ \cot(a + b) = \frac{\cot a \cot b - 1}{\cot a + \cot b} \]
In our case, let \( a = \theta \) and \( b = 4\theta \). Then:
\[ \cot(5\theta) = \frac{\cot \theta \cot 4\theta - 1}{\cot \theta + \cot 4\theta} \]
Thus, we can substitute this into our expression for \( \cot 7\theta \):
\[ \cot 7\theta = -\cot 5\theta \]

### Step 4: Setting Up the Equation
Now we have:
\[ \cot 5\theta + \cot 7\theta = 0 \]
This implies:
\[ \cot 5\theta = -\cot 7\theta \]

### Step 5: Using the Cotangent Identity
Using the identity \( \cot x = \frac{\cos x}{\sin x} \):
\[ \frac{\cos 5\theta}{\sin 5\theta} + \frac{\cos 7\theta}{\sin 7\theta} = 0 \]
This leads to:
\[ \cos 5\theta \sin 7\theta + \sin 5\theta \cos 7\theta = 0 \]

### Step 6: Applying the Sine Addition Formula
This can be rewritten using the sine addition formula:
\[ \sin(5\theta + 7\theta) = \sin(12\theta) = 0 \]

### Step 7: Finding the General Solution
The general solution for \( \sin x = 0 \) is:
\[ 12\theta = n\pi \quad \text{where } n \in \mathbb{Z} \]
Thus, we can solve for \( \theta \):
\[ \theta = \frac{n\pi}{12} \]

### Final Answer
The values of \( \theta \) that satisfy the original equation are:
\[ \theta = \frac{n\pi}{12}, \quad n \in \mathbb{Z} \]