If `k=(1-sinalpha).(1-sinbeta).(1-singamma)=(1+sinalpha).(1+sinbeta).(1+singamma)` then find k.

Answer

To solve the equation \( k = (1 - \sin \alpha)(1 - \sin \beta)(1 - \sin \gamma) = (1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma) \), we can follow these steps:

### Step 1: Set up the equation
We start with the equation given in the problem:
\[ k = (1 - \sin \alpha)(1 - \sin \beta)(1 - \sin \gamma) = (1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma) \]

### Step 2: Multiply both sides
We can multiply both sides of the equation:
\[ k \cdot k = (1 - \sin \alpha)(1 + \sin \alpha)(1 - \sin \beta)(1 + \sin \beta)(1 - \sin \gamma)(1 + \sin \gamma) \]

### Step 3: Apply the difference of squares
Using the identity \( (a - b)(a + b) = a^2 - b^2 \), we can simplify:
\[ k^2 = (1 - \sin^2 \alpha)(1 - \sin^2 \beta)(1 - \sin^2 \gamma) \]

### Step 4: Use the Pythagorean identity
We know that \( 1 - \sin^2 \theta = \cos^2 \theta \). Therefore, we can rewrite the equation:
\[ k^2 = \cos^2 \alpha \cdot \cos^2 \beta \cdot \cos^2 \gamma \]

### Step 5: Take the square root
To find \( k \), we take the square root of both sides:
\[ k = \sqrt{\cos^2 \alpha \cdot \cos^2 \beta \cdot \cos^2 \gamma} \]
This simplifies to:
\[ k = |\cos \alpha \cdot \cos \beta \cdot \cos \gamma| \]

### Conclusion
Thus, the final result for \( k \) is:
\[ k = \cos \alpha \cdot \cos \beta \cdot \cos \gamma \quad \text{(considering positive values)} \]