If : `(1-sin alpha)(1-sin beta)(1-singamma)=(1+sinalpha)(1+sin beta)(1+sin gamma),`
then one value of each side is

Options

A. `cos alpha* cos beta* cos gamma`
B. `sin alpha* sin beta* sin gamma`
C. `sec alpha* sec beta* sec gamma`
D. `tan alpha* tan beta* tan gamma`

Answer

To solve the equation \((1 - \sin \alpha)(1 - \sin \beta)(1 - \sin \gamma) = (1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma)\), we will manipulate both sides step by step.

### Step 1: Rewrite the equation
We start with the given equation:
\[ (1 - \sin \alpha)(1 - \sin \beta)(1 - \sin \gamma) = (1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma) \]

### Step 2: Use the identity \(1 - \sin^2 x = \cos^2 x\)
We can rewrite the left-hand side (LHS) using the identity \(1 - \sin^2 x = \cos^2 x\):
\[ (1 - \sin \alpha)(1 - \sin \beta)(1 - \sin \gamma) = \frac{(1 - \sin \alpha)(1 - \sin \beta)(1 - \sin \gamma)(1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma)}{(1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma)} \]

### Step 3: Simplify the LHS
Now we can express the LHS as:
\[ (1 - \sin^2 \alpha)(1 - \sin^2 \beta)(1 - \sin^2 \gamma) = \cos^2 \alpha \cos^2 \beta \cos^2 \gamma \]

### Step 4: Rewrite the RHS
The right-hand side (RHS) can be expressed similarly:
\[ (1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma) \]

### Step 5: Set the two sides equal
Now we have:
\[ \frac{\cos^2 \alpha \cos^2 \beta \cos^2 \gamma}{(1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma)} = (1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma) \]

### Step 6: Cross-multiply
Cross-multiplying gives us:
\[ \cos^2 \alpha \cos^2 \beta \cos^2 \gamma = (1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma)(1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma) \]

### Step 7: Final expression
Thus, we find that:
\[ \cos^2 \alpha \cos^2 \beta \cos^2 \gamma = (1 + \sin \alpha)(1 + \sin \beta)(1 + \sin \gamma) \]

### Conclusion
From the above steps, we conclude that the equation holds true under the condition that:
\[ \cos \alpha \cos \beta \cos \gamma = 1 + \sin \alpha + \sin \beta + \sin \gamma \]