The value of `tan10^(@)tan15^(@)tan75^(@)tan80^(@)` is

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the problem of finding the value of \( \tan(10^\circ) \tan(15^\circ) \tan(75^\circ) \tan(80^\circ) \), we can use the properties of trigonometric functions.

### Step-by-Step Solution:

1. **Identify the relationships**:
We know that:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
This means:
\[ \tan(75^\circ) = \cot(15^\circ) \quad \text{and} \quad \tan(80^\circ) = \cot(10^\circ) \]

2. **Rewrite the expression**:
Using the relationships identified, we can rewrite the expression:
\[ \tan(10^\circ) \tan(15^\circ) \tan(75^\circ) \tan(80^\circ = \tan(10^\circ) \tan(15^\circ) \cot(15^\circ) \cot(10^\circ) \]

3. **Use the identity**:
We know that:
\[ \tan(\theta) \cot(\theta) = 1 \]
Therefore, we can simplify the expression:
\[ \tan(10^\circ) \cot(10^\circ) = 1 \quad \text{and} \quad \tan(15^\circ) \cot(15^\circ) = 1 \]

4. **Combine the results**:
Thus, we have:
\[ \tan(10^\circ) \tan(15^\circ) \cot(15^\circ) \cot(10^\circ) = 1 \cdot 1 = 1 \]

5. **Final result**:
Therefore, the value of \( \tan(10^\circ) \tan(15^\circ) \tan(75^\circ) \tan(80^\circ) \) is:
\[ \boxed{1} \]