If tan `theta tan 5theta = 1`, then what is the value of sin `2theta`?

Options

A. 0
B. `1//2`
C. `1sqrt()2`
D. `sqrt()3//2`

Answer

To solve the equation \( \tan \theta \tan 5\theta = 1 \) and find the value of \( \sin 2\theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation**:
\[ \tan \theta \tan 5\theta = 1 \]

2. **Rearrange the equation**:
We can express \( \tan 5\theta \) in terms of \( \tan \theta \):
\[ \tan 5\theta = \frac{1}{\tan \theta} \]

3. **Use the identity for tangent**:
We know that:
\[ \tan 5\theta = \tan(90^\circ - \theta) \]
This implies:
\[ 5\theta = 90^\circ - \theta \]

4. **Solve for \( \theta \)**:
Rearranging gives:
\[ 5\theta + \theta = 90^\circ \]
\[ 6\theta = 90^\circ \]
\[ \theta = 15^\circ \]

5. **Find \( 2\theta \)**:
Now, calculate \( 2\theta \):
\[ 2\theta = 2 \times 15^\circ = 30^\circ \]

6. **Calculate \( \sin 2\theta \)**:
We now find \( \sin 2\theta \):
\[ \sin 2\theta = \sin 30^\circ = \frac{1}{2} \]

### Final Answer:
Thus, the value of \( \sin 2\theta \) is:
\[ \frac{1}{2} \]