(1+sinA)(1+sinB)(1+sinC) = (1−sinA)(1−sinB)(1−sinC) =k ⇒ k =

1. Equate the two expressions:

(1 + sinA)(1 + sinB)(1 + sinC) = (1 - sinA)(1 - sinB)(1 - sinC)

2. Multiply both sides:

(1 + sinA)(1 + sinB)(1 + sinC) × (1 - sinA)(1 - sinB)(1 - sinC) = (1 - sinA)(1 - sinB)(1 - sinC) × (1 + sinA)(1 + sinB)(1 + sinC)

3. Simplify each term:

(1 - sin²A)(1 - sin²B)(1 - sin²C) = cos²A ⋅ cos²B ⋅ cos²C

4. Relate to k:

Since the original equation states that both expressions are equal to k:

k² = cos²A ⋅ cos²B ⋅ cos²C

5. Solve for k:

k = ± (cosA ⋅ cosB ⋅ cosC)