Evaluate without using trigonometric tables :
`tan 20^(@) tan 40^(@) tan 50^(@) tan 70^(@)`

Answer

To evaluate the expression \( \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ \) without using trigonometric tables, we can use the properties of trigonometric functions.

### Step-by-Step Solution:

1. **Identify Relationships**:
We know that:
\[ \tan(90^\circ - x) = \cot x \]
This means that:
\[ \tan 70^\circ = \cot 20^\circ \quad \text{and} \quad \tan 50^\circ = \cot 40^\circ \]

2. **Rewrite the Expression**:
Using the relationships identified, we can rewrite the expression:
\[ \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ = \tan 20^\circ \tan 40^\circ \cot 40^\circ \cot 20^\circ \]

3. **Use the Cotangent Identity**:
Recall that:
\[ \cot x = \frac{1}{\tan x} \]
Therefore:
\[ \cot 40^\circ = \frac{1}{\tan 40^\circ} \quad \text{and} \quad \cot 20^\circ = \frac{1}{\tan 20^\circ} \]

4. **Substitute Cotangent Values**:
Substitute the cotangent values into the expression:
\[ = \tan 20^\circ \tan 40^\circ \cdot \frac{1}{\tan 40^\circ} \cdot \frac{1}{\tan 20^\circ} \]

5. **Simplify the Expression**:
Now, we can simplify:
\[ = \tan 20^\circ \cdot \frac{1}{\tan 20^\circ} \cdot \tan 40^\circ \cdot \frac{1}{\tan 40^\circ} = 1 \cdot 1 = 1 \]

### Final Result:
Thus, the value of \( \tan 20^\circ \tan 40^\circ \tan 50^\circ \tan 70^\circ \) is:
\[ \boxed{1} \]