With out using tables, evaluate :
`4 tan 60^(@) sec 30^(@) + (sin 31^(@) sec 59^(@) + cot 59^(@) cot 31^(@))/(8 sin^(2) 30^(@) - tan^(2) 45^(@))`

Answer

To evaluate the expression \( 4 \tan 60^\circ \sec 30^\circ + \frac{\sin 31^\circ \sec 59^\circ + \cot 59^\circ \cot 31^\circ}{8 \sin^2 30^\circ - \tan^2 45^\circ} \), we will break it down step by step.

### Step 1: Evaluate \( 4 \tan 60^\circ \sec 30^\circ \)

1. **Calculate \( \tan 60^\circ \)**:
\[ \tan 60^\circ = \sqrt{3} \]

2. **Calculate \( \sec 30^\circ \)**:
\[ \sec 30^\circ = \frac{1}{\cos 30^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} \]

3. **Combine the results**:
\[ 4 \tan 60^\circ \sec 30^\circ = 4 \cdot \sqrt{3} \cdot \frac{2}{\sqrt{3}} = 4 \cdot 2 = 8 \]

### Step 2: Evaluate the denominator \( 8 \sin^2 30^\circ - \tan^2 45^\circ \)

1. **Calculate \( \sin 30^\circ \)**:
\[ \sin 30^\circ = \frac{1}{2} \]

2. **Calculate \( \sin^2 30^\circ \)**:
\[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \]

3. **Calculate \( \tan 45^\circ \)**:
\[ \tan 45^\circ = 1 \]

4. **Calculate \( \tan^2 45^\circ \)**:
\[ \tan^2 45^\circ = 1^2 = 1 \]

5. **Combine the results**:
\[ 8 \sin^2 30^\circ - \tan^2 45^\circ = 8 \cdot \frac{1}{4} - 1 = 2 - 1 = 1 \]

### Step 3: Evaluate \( \sin 31^\circ \sec 59^\circ + \cot 59^\circ \cot 31^\circ \)

1. **Using the identity \( \sec(90^\circ - \theta) = \csc \theta \)**:
\[ \sec 59^\circ = \csc 31^\circ \]

2. **Thus, \( \sin 31^\circ \sec 59^\circ = \sin 31^\circ \cdot \csc 31^\circ = 1 \)**.

3. **Using the identity \( \cot(90^\circ - \theta) = \tan \theta \)**:
\[ \cot 59^\circ = \tan 31^\circ \]

4. **Thus, \( \cot 59^\circ \cot 31^\circ = \tan 31^\circ \cdot \cot 31^\circ = 1 \)**.

5. **Combine the results**:
\[ \sin 31^\circ \sec 59^\circ + \cot 59^\circ \cot 31^\circ = 1 + 1 = 2 \]

### Step 4: Combine everything

Now we can put everything together:
\[ 4 \tan 60^\circ \sec 30^\circ + \frac{2}{1} = 8 + 2 = 10 \]

### Final Answer
\[ \text{The final answer is } 10. \]