`cos^(2) alpha +cos^(2) beta +cos^(2) gamma ` is equal to

Answer

To solve the problem of finding the value of \( \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma \), we can use the geometric interpretation of angles made by a vector with the coordinate axes.

### Step-by-Step Solution:

1. **Understanding the Angles**:
- Let \( \alpha \), \( \beta \), and \( \gamma \) be the angles that a vector makes with the x-axis, y-axis, and z-axis respectively.

2. **Using the Cosine Definition**:
- The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. In three-dimensional space, for a unit vector, the cosines of the angles with the axes represent the projections of the vector onto those axes.

3. **Applying the Pythagorean Identity**:
- For any vector in three-dimensional space, the sum of the squares of the cosines of the angles with the coordinate axes is equal to 1. This is derived from the Pythagorean theorem.
- Mathematically, this can be expressed as:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

4. **Conclusion**:
- Therefore, we conclude that:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]

### Final Answer:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]