If `tan 3 theta. tan 7 theta. = 1`, then the value of `tan (theta + 36^(@))` is :

Options

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

Answer

To solve the problem, we need to find the value of \( \tan(\theta + 36^\circ) \) given that \( \tan(3\theta) \cdot \tan(7\theta) = 1 \).

### Step-by-Step Solution:

1. **Understanding the Equation**:
We start with the equation:
\[ \tan(3\theta) \cdot \tan(7\theta) = 1 \]
This implies:
\[ \tan(7\theta) = \frac{1}{\tan(3\theta)} = \cot(3\theta) \]

2. **Using the Cotangent Identity**:
From the identity \( \cot(x) = \tan(90^\circ - x) \), we can rewrite \( \tan(7\theta) \):
\[ \tan(7\theta) = \tan(90^\circ - 3\theta) \]

3. **Setting Up the Angle Equation**:
Since \( \tan(7\theta) = \tan(90^\circ - 3\theta) \), we can equate the angles:
\[ 7\theta = 90^\circ - 3\theta + n \cdot 180^\circ \quad \text{(for some integer } n\text{)} \]
Simplifying gives:
\[ 10\theta = 90^\circ + n \cdot 180^\circ \]
Thus:
\[ \theta = 9^\circ + n \cdot 18^\circ \]

4. **Finding \( \tan(\theta + 36^\circ) \)**:
We need to find \( \tan(\theta + 36^\circ) \). Substituting \( \theta = 9^\circ \):
\[ \tan(9^\circ + 36^\circ) = \tan(45^\circ) \]

5. **Calculating the Final Value**:
We know that:
\[ \tan(45^\circ) = 1 \]

### Conclusion:
Thus, the value of \( \tan(\theta + 36^\circ) \) is:
\[ \boxed{1} \]