If `(l_(1), m_(1), n_(1)) , (l_(2), m_(2), n_(2))` are D.C's of two lines, then `(l_(1)m_(2)-l_(2)m_(1))^2+(m_(1)n_(2)-n_(1)m_(2))^2+(n_(1)l_(2)-n_(2)l_(1))^2+(l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2))^2=`

Answer

To solve the problem, we need to evaluate the expression:

\[ (l_1 m_2 - l_2 m_1)^2 + (m_1 n_2 - n_1 m_2)^2 + (n_1 l_2 - n_2 l_1)^2 + (l_1 l_2 + m_1 m_2 + n_1 n_2)^2 \]

where \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) are the direction cosines of two lines.

### Step 1: Understand the properties of direction cosines

Since \((l_1, m_1, n_1)\) and \((l_2, m_2, n_2)\) are direction cosines, we know that:

\[ l_1^2 + m_1^2 + n_1^2 = 1 \]
\[ l_2^2 + m_2^2 + n_2^2 = 1 \]

### Step 2: Expand the expression

We will expand each term in the expression.

1. **First term**:
\[ (l_1 m_2 - l_2 m_1)^2 = l_1^2 m_2^2 + l_2^2 m_1^2 - 2 l_1 l_2 m_1 m_2 \]

2. **Second term**:
\[ (m_1 n_2 - n_1 m_2)^2 = m_1^2 n_2^2 + n_1^2 m_2^2 - 2 m_1 n_1 m_2 n_2 \]

3. **Third term**:
\[ (n_1 l_2 - n_2 l_1)^2 = n_1^2 l_2^2 + n_2^2 l_1^2 - 2 n_1 n_2 l_1 l_2 \]

4. **Fourth term**:
\[ (l_1 l_2 + m_1 m_2 + n_1 n_2)^2 = l_1^2 l_2^2 + m_1^2 m_2^2 + n_1^2 n_2^2 + 2(l_1 l_2 m_1 m_2 + m_1 m_2 n_1 n_2 + n_1 n_2 l_1 l_2) \]

### Step 3: Combine all terms

Now, we will combine all the expanded terms:

\[ = (l_1^2 m_2^2 + l_2^2 m_1^2 - 2 l_1 l_2 m_1 m_2) + (m_1^2 n_2^2 + n_1^2 m_2^2 - 2 m_1 n_1 m_2 n_2) + (n_1^2 l_2^2 + n_2^2 l_1^2 - 2 n_1 n_2 l_1 l_2) + (l_1^2 l_2^2 + m_1^2 m_2^2 + n_1^2 n_2^2 + 2(l_1 l_2 m_1 m_2 + m_1 m_2 n_1 n_2 + n_1 n_2 l_1 l_2)) \]

### Step 4: Simplify the expression

Now, we can simplify the expression by combining like terms:

- The terms with negative coefficients will cancel out with the positive coefficients from the fourth term.
- This leads to:

\[ = l_1^2 (m_2^2 + l_2^2) + m_1^2 (n_2^2 + l_2^2) + n_1^2 (l_2^2 + m_2^2) \]

### Step 5: Use the properties of direction cosines

Since \(l_2^2 + m_2^2 + n_2^2 = 1\) and \(l_1^2 + m_1^2 + n_1^2 = 1\), we can substitute:

\[ = l_1^2 \cdot 1 + m_1^2 \cdot 1 + n_1^2 \cdot 1 = l_1^2 + m_1^2 + n_1^2 = 1 \]

### Final Result

Thus, the value of the expression is:

\[ \boxed{1} \]