Evaluate : `(3tan25^(@)tan40^(@)tan50^(@)tan65^(@)-(1)/(2)tan^(2)60^(@))/(4(cos^(2)29^(@)+cos^(2)61^(2)))`

Answer

To evaluate the expression:

\[ \frac{3 \tan 25^\circ \tan 40^\circ \tan 50^\circ \tan 65^\circ - \frac{1}{2} \tan^2 60^\circ}{4(\cos^2 29^\circ + \cos^2 61^\circ)} \]

we will follow these steps:

### Step 1: Simplify the tangent terms
Using the identity \( \tan(90^\circ - \theta) = \cot(\theta) \), we can rewrite the tangent terms:

- \( \tan 65^\circ = \cot 25^\circ \)
- \( \tan 50^\circ = \cot 40^\circ \)

Thus, we can rewrite the expression as:

\[ 3 \tan 25^\circ \cot 25^\circ \tan 40^\circ \cot 40^\circ \]

Since \( \tan \theta \cot \theta = 1 \), we have:

\[ 3 \cdot 1 \cdot 1 = 3 \]

So the numerator becomes:

\[ 3 - \frac{1}{2} \tan^2 60^\circ \]

### Step 2: Calculate \( \tan 60^\circ \)
We know that:

\[ \tan 60^\circ = \sqrt{3} \]

Thus, \( \tan^2 60^\circ = 3 \). Therefore, we substitute this into the numerator:

\[ 3 - \frac{1}{2} \cdot 3 = 3 - \frac{3}{2} = \frac{6}{2} - \frac{3}{2} = \frac{3}{2} \]

### Step 3: Simplify the cosine terms
Now, let's simplify the denominator:

\[ 4(\cos^2 29^\circ + \cos^2 61^\circ) \]

Using the identity \( \cos(90^\circ - \theta) = \sin(\theta) \), we have:

\[ \cos 61^\circ = \sin 29^\circ \]

Thus, we can rewrite the denominator as:

\[ 4(\cos^2 29^\circ + \sin^2 29^\circ) \]

Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ 4(1) = 4 \]

### Step 4: Combine the results
Now we can combine the results from the numerator and denominator:

\[ \frac{\frac{3}{2}}{4} = \frac{3}{2 \cdot 4} = \frac{3}{8} \]

### Final Answer
Thus, the final answer is:

\[ \frac{3}{8} \]