If `tan3theta.tan7theta=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 where \( \tan 3\theta \cdot \tan 7\theta = 1 \) and find the value of \( \tan(\theta + 36^\circ) \), we can follow these steps:

### Step-by-step Solution:

1. **Use the Identity**: We know that if \( \tan A \cdot \tan B = 1 \), then \( A + B = 90^\circ \). Here, let \( A = 3\theta \) and \( B = 7\theta \).
\[ 3\theta + 7\theta = 90^\circ \]

2. **Combine the Angles**: Combine the angles from the equation above.
\[ 10\theta = 90^\circ \]

3. **Solve for \( \theta \)**: Divide both sides by 10 to find \( \theta \).
\[ \theta = \frac{90^\circ}{10} = 9^\circ \]

4. **Find \( \tan(\theta + 36^\circ) \)**: Now substitute \( \theta \) into \( \tan(\theta + 36^\circ) \).
\[ \tan(9^\circ + 36^\circ) = \tan(45^\circ) \]

5. **Evaluate \( \tan(45^\circ) \)**: We know that \( \tan(45^\circ) = 1 \).
\[ \tan(9^\circ + 36^\circ) = 1 \]

### Final Answer:
The value of \( \tan(\theta + 36^\circ) \) is \( 1 \).

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