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 given expression \((l_{1}m_{2}-l_{2}m_{1})^2+(m_{1}n_{2}-n_{1}m_{2})^2+(n_{1}l_{2}-n_{2}l_{1})^2+(l_{1}l_{2}+m_{1}m_{2}+n_{1}n_{2})^2\), we will follow these steps:

### Step 1: Understand the Given Information
We are given two lines with direction cosines \((l_{1}, m_{1}, n_{1})\) and \((l_{2}, m_{2}, n_{2})\). The direction cosines satisfy the condition:
\[ l_{1}^2 + m_{1}^2 + n_{1}^2 = 1 \]
\[ l_{2}^2 + m_{2}^2 + n_{2}^2 = 1 \]

### Step 2: Expand Each Term
We will expand each of the squared terms in the expression.

1. **First Term**:
\[ (l_{1}m_{2} - l_{2}m_{1})^2 = l_{1}^2m_{2}^2 + l_{2}^2m_{1}^2 - 2l_{1}m_{2}l_{2}m_{1} \]

2. **Second Term**:
\[ (m_{1}n_{2} - n_{1}m_{2})^2 = m_{1}^2n_{2}^2 + n_{1}^2m_{2}^2 - 2m_{1}n_{2}n_{1}m_{2} \]

3. **Third Term**:
\[ (n_{1}l_{2} - n_{2}l_{1})^2 = n_{1}^2l_{2}^2 + n_{2}^2l_{1}^2 - 2n_{1}l_{2}n_{2}l_{1} \]

4. **Fourth Term**:
\[ (l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2})^2 = (l_{1}l_{2})^2 + (m_{1}m_{2})^2 + (n_{1}n_{2})^2 + 2(l_{1}l_{2}m_{1}m_{2} + m_{1}m_{2}n_{1}n_{2} + n_{1}n_{2}l_{1}l_{2}) \]

### Step 3: Combine All Terms
Now, we will combine all the expanded terms:
\[ = (l_{1}^2m_{2}^2 + l_{2}^2m_{1}^2 - 2l_{1}m_{2}l_{2}m_{1}) + (m_{1}^2n_{2}^2 + n_{1}^2m_{2}^2 - 2m_{1}n_{2}n_{1}m_{2}) + (n_{1}^2l_{2}^2 + n_{2}^2l_{1}^2 - 2n_{1}l_{2}n_{2}l_{1}) + (l_{1}l_{2})^2 + (m_{1}m_{2})^2 + (n_{1}n_{2})^2 + 2(l_{1}l_{2}m_{1}m_{2} + m_{1}m_{2}n_{1}n_{2} + n_{1}n_{2}l_{1}l_{2}) \]

### Step 4: Simplify the Expression
Now, we will simplify the expression. Notice that:
- The terms involving products of direction cosines will cancel out due to symmetry and the properties of direction cosines.

After simplification, we find that:
\[ = 1 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{1} \]