If `alpha, beta and gamma` are angles made by the line with positive direction of X-axis, Y-axis and Z-axis respectively, then find the value of `cos2alpha+cos2beta+cos2gamma`.

Answer

To solve the problem, we need to find the value of \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma \) where \( \alpha, \beta, \) and \( \gamma \) are the angles made by a line with the positive directions of the X-axis, Y-axis, and Z-axis, respectively.

### Step-by-Step Solution:

1. **Understanding Direction Cosines**:
The direction cosines of a line making angles \( \alpha, \beta, \) and \( \gamma \) with the coordinate axes are given by:
\[ l = \cos \alpha, \quad m = \cos \beta, \quad n = \cos \gamma \]

2. **Using the Relationship of Direction Cosines**:
The relationship between the direction cosines is given by:
\[ l^2 + m^2 + n^2 = 1 \]
Substituting the values of \( l, m, n \):
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

3. **Using the Cosine Double Angle Identity**:
We know the double angle identity for cosine:
\[ \cos 2\theta = 2\cos^2 \theta - 1 \]
Therefore, we can express \( \cos^2 \alpha, \cos^2 \beta, \) and \( \cos^2 \gamma \) in terms of \( \cos 2\alpha, \cos 2\beta, \) and \( \cos 2\gamma \):
\[ \cos^2 \alpha = \frac{1 + \cos 2\alpha}{2}, \quad \cos^2 \beta = \frac{1 + \cos 2\beta}{2}, \quad \cos^2 \gamma = \frac{1 + \cos 2\gamma}{2} \]

4. **Substituting into the Equation**:
Substitute these expressions into the equation \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \):
\[ \frac{1 + \cos 2\alpha}{2} + \frac{1 + \cos 2\beta}{2} + \frac{1 + \cos 2\gamma}{2} = 1 \]

5. **Simplifying the Equation**:
Combine the terms:
\[ \frac{3 + \cos 2\alpha + \cos 2\beta + \cos 2\gamma}{2} = 1 \]
Multiply both sides by 2:
\[ 3 + \cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2 \]

6. **Isolating the Cosine Terms**:
Rearranging gives:
\[ \cos 2\alpha + \cos 2\beta + \cos 2\gamma = 2 - 3 = -1 \]

### Final Result:
Thus, the value of \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma \) is:
\[ \boxed{-1} \]