If `cosalpha, cosbeta and cosgamma` are the direction cosine of a line, then find the value of `cos^2alpha+(cosbeta+singamma)(cosbeta-singamma)`.

Answer

To solve the problem, we need to find the value of the expression \( \cos^2 \alpha + (\cos \beta + \sin \gamma)(\cos \beta - \sin \gamma) \).

### Step-by-Step Solution:

1. **Understand the Direction Cosines**:
The direction cosines of a line are given as \( \cos \alpha, \cos \beta, \cos \gamma \). By definition, the sum of the squares of the direction cosines equals 1:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

2. **Simplify the Expression**:
We can rewrite the expression \( \cos^2 \alpha + (\cos \beta + \sin \gamma)(\cos \beta - \sin \gamma) \) using the difference of squares:
\[ (\cos \beta + \sin \gamma)(\cos \beta - \sin \gamma) = \cos^2 \beta - \sin^2 \gamma \]
Therefore, the expression becomes:
\[ \cos^2 \alpha + \cos^2 \beta - \sin^2 \gamma \]

3. **Substituting for \( \sin^2 \gamma \)**:
We know from the Pythagorean identity that:
\[ \sin^2 \gamma = 1 - \cos^2 \gamma \]
Substituting this into our expression gives:
\[ \cos^2 \alpha + \cos^2 \beta - (1 - \cos^2 \gamma) \]
Simplifying this, we get:
\[ \cos^2 \alpha + \cos^2 \beta - 1 + \cos^2 \gamma \]

4. **Rearranging the Expression**:
Now, we can rearrange the terms:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma - 1 \]

5. **Using the Property of Direction Cosines**:
From the property of direction cosines, we have:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
Substituting this into our expression gives:
\[ 1 - 1 = 0 \]

6. **Final Result**:
Therefore, the value of the expression \( \cos^2 \alpha + (\cos \beta + \sin \gamma)(\cos \beta - \sin \gamma) \) is:
\[ \boxed{0} \]