A line makes an angle `alpha, beta, gamma, delta` with the four diagonals of a cube, then `sin^(2)alpha+sin^(2)beta+sin^(2)gamma+sin^(2)delta=`

Options

A. `(4)/(3)`
B. `(8)/(3)`
C. `(1)/(3)`
D. `(2)/(3)`

Answer

To solve the problem, we need to find the value of \( \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma + \sin^2 \delta \), where \( \alpha, \beta, \gamma, \delta \) are the angles made by a line with the four diagonals of a cube.

### Step-by-Step Solution:

1. **Understanding the Angles**:
The angles \( \alpha, \beta, \gamma, \delta \) correspond to the angles that a line makes with the four space diagonals of a cube.

2. **Using the Property of Cosine**:
We know from geometry that for any angles \( \alpha, \beta, \gamma, \delta \), the following relationship holds:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma + \cos^2 \delta = \frac{4}{3} \]
This is a known result for angles made with the diagonals of a cube.

3. **Relating Sine and Cosine**:
We can use the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) to express the cosine terms in terms of sine:
\[ \cos^2 \alpha = 1 - \sin^2 \alpha \]
\[ \cos^2 \beta = 1 - \sin^2 \beta \]
\[ \cos^2 \gamma = 1 - \sin^2 \gamma \]
\[ \cos^2 \delta = 1 - \sin^2 \delta \]

4. **Substituting into the Cosine Property**:
Substitute these expressions into the cosine property:
\[ (1 - \sin^2 \alpha) + (1 - \sin^2 \beta) + (1 - \sin^2 \gamma) + (1 - \sin^2 \delta) = \frac{4}{3} \]

5. **Simplifying the Equation**:
This simplifies to:
\[ 4 - (\sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma + \sin^2 \delta) = \frac{4}{3} \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma + \sin^2 \delta = 4 - \frac{4}{3} \]

7. **Calculating the Right Side**:
To calculate \( 4 - \frac{4}{3} \):
\[ 4 = \frac{12}{3} \quad \text{(converting 4 to a fraction with denominator 3)} \]
Therefore:
\[ 4 - \frac{4}{3} = \frac{12}{3} - \frac{4}{3} = \frac{8}{3} \]

8. **Final Result**:
Thus, we find that:
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma + \sin^2 \delta = \frac{8}{3} \]

### Conclusion:
The value of \( \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma + \sin^2 \delta \) is \( \frac{8}{3} \).