Evaluate `tan20^(@)tan25^(@)tan65^(@)tan70^(@)`.

Answer

To evaluate \( \tan 20^\circ \tan 25^\circ \tan 65^\circ \tan 70^\circ \), we can use the properties of trigonometric functions. Here’s a step-by-step solution:

### Step 1: Group the terms
We can rearrange the expression:
\[ \tan 20^\circ \tan 70^\circ \tan 25^\circ \tan 65^\circ \]

### Step 2: Use the complementary angle identity
Recall that:
\[ \tan(90^\circ - \theta) = \cot \theta \]
Using this identity, we can rewrite the terms:
\[ \tan 70^\circ = \cot 20^\circ \quad \text{and} \quad \tan 65^\circ = \cot 25^\circ \]
Thus, we can substitute:
\[ \tan 20^\circ \tan 70^\circ = \tan 20^\circ \cot 20^\circ \]
\[ \tan 25^\circ \tan 65^\circ = \tan 25^\circ \cot 25^\circ \]

### Step 3: Simplify using the identity
Using the identity \( \tan \theta \cot \theta = 1 \):
\[ \tan 20^\circ \cot 20^\circ = 1 \quad \text{and} \quad \tan 25^\circ \cot 25^\circ = 1 \]
So we have:
\[ \tan 20^\circ \tan 70^\circ \tan 25^\circ \tan 65^\circ = 1 \cdot 1 = 1 \]

### Conclusion
Thus, the value of \( \tan 20^\circ \tan 25^\circ \tan 65^\circ \tan 70^\circ \) is:
\[ \boxed{1} \]