If ` tan8 theta = cot 2 theta`, where `0 lt 8 theta lt pi/2`, then what is the value of `tan 5 theta` ?

Options

A. `1/sqrt3`
B. 1
C. `sqrt3`
D. 0

Answer

To solve the equation \( \tan(8\theta) = \cot(2\theta) \) and find the value of \( \tan(5\theta) \), we can follow these steps:

### Step 1: Rewrite the cotangent in terms of tangent
We know that \( \cot(x) = \frac{1}{\tan(x)} \). Therefore, we can rewrite the equation as:
\[ \tan(8\theta) = \frac{1}{\tan(2\theta)} \]

### Step 2: Cross-multiply
Cross-multiplying gives us:
\[ \tan(8\theta) \cdot \tan(2\theta) = 1 \]

### Step 3: Use the identity for tangent
From the identity \( \tan(A) \tan(B) = 1 \) when \( A + B = \frac{\pi}{2} \), we can set:
\[ 8\theta + 2\theta = \frac{\pi}{2} \]
This simplifies to:
\[ 10\theta = \frac{\pi}{2} \]

### Step 4: Solve for \( \theta \)
Dividing both sides by 10:
\[ \theta = \frac{\pi}{20} \]

### Step 5: Find \( \tan(5\theta) \)
Now, we need to find \( \tan(5\theta) \):
\[ 5\theta = 5 \cdot \frac{\pi}{20} = \frac{\pi}{4} \]
Thus:
\[ \tan(5\theta) = \tan\left(\frac{\pi}{4}\right) = 1 \]

### Conclusion
The value of \( \tan(5\theta) \) is \( 1 \).