Evaluate :
` tan 25^(@) tan 65 ^(@) - cot 25^(@) cot 65^(@) `

Answer

To evaluate the expression \( \tan 25^\circ \tan 65^\circ - \cot 25^\circ \cot 65^\circ \), we can use trigonometric identities. Here’s a step-by-step solution:

### Step 1: Rewrite the angles
We know that:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
Thus, we can rewrite \( \tan 65^\circ \) as:
\[ \tan 65^\circ = \tan(90^\circ - 25^\circ) = \cot 25^\circ \]

### Step 2: Substitute into the expression
Now substituting \( \tan 65^\circ \) in the original expression, we get:
\[ \tan 25^\circ \tan 65^\circ = \tan 25^\circ \cot 25^\circ \]

### Step 3: Simplify \( \tan 25^\circ \cot 25^\circ \)
Since \( \tan \theta \cdot \cot \theta = 1 \), we have:
\[ \tan 25^\circ \cot 25^\circ = 1 \]

### Step 4: Rewrite \( \cot 65^\circ \)
Similarly, we can rewrite \( \cot 65^\circ \):
\[ \cot 65^\circ = \cot(90^\circ - 25^\circ) = \tan 25^\circ \]

### Step 5: Substitute \( \cot 65^\circ \) into the expression
Now substituting \( \cot 65^\circ \) in the original expression, we have:
\[ \tan 25^\circ \tan 65^\circ - \cot 25^\circ \cot 65^\circ = 1 - \cot 25^\circ \tan 25^\circ \]

### Step 6: Simplify \( \cot 25^\circ \tan 25^\circ \)
Again, using the identity \( \cot \theta \cdot \tan \theta = 1 \):
\[ \cot 25^\circ \tan 25^\circ = 1 \]

### Step 7: Final simplification
Now, substituting back into the expression:
\[ 1 - 1 = 0 \]

### Conclusion
Thus, the value of the expression \( \tan 25^\circ \tan 65^\circ - \cot 25^\circ \cot 65^\circ \) is:
\[ \boxed{0} \]