If `x=sin(alpha-beta)*sin(gamma-delta), y=sin(beta-gamma)*sin(alpha-delta), z=sin(gamma-alpha) *sin(beta-delta)`, then :

Options

A. `x+y+z=0`
B. `x^(3)+y^(3)+z^(3)=3xyz`
C. `x+y-z=0`
D. `x^(3)+y^(3)-z^(3)=3xyz`

Answer

To solve the problem step by step, we will analyze the expressions for \( x \), \( y \), and \( z \) given in the question.

### Step 1: Express \( x \)
Given:
\[ x = \sin(\alpha - \beta) \cdot \sin(\gamma - \delta) \]
Using the sine difference identity:
\[ \sin(A - B) = \sin A \cos B - \cos A \sin B \]
we can write:
\[ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \]
and
\[ \sin(\gamma - \delta) = \sin \gamma \cos \delta - \cos \gamma \sin \delta \]
Substituting these into \( x \):
\[ x = (\sin \alpha \cos \beta - \cos \alpha \sin \beta)(\sin \gamma \cos \delta - \cos \gamma \sin \delta) \]
Now, expanding this product:
\[ x = \sin \alpha \cos \beta \sin \gamma \cos \delta - \sin \alpha \cos \beta \cos \gamma \sin \delta - \cos \alpha \sin \beta \sin \gamma \cos \delta + \cos \alpha \sin \beta \cos \gamma \sin \delta \]

### Step 2: Express \( y \)
Given:
\[ y = \sin(\beta - \gamma) \cdot \sin(\alpha - \delta) \]
Using the sine difference identity again:
\[ \sin(\beta - \gamma) = \sin \beta \cos \gamma - \cos \beta \sin \gamma \]
and
\[ \sin(\alpha - \delta) = \sin \alpha \cos \delta - \cos \alpha \sin \delta \]
Substituting these into \( y \):
\[ y = (\sin \beta \cos \gamma - \cos \beta \sin \gamma)(\sin \alpha \cos \delta - \cos \alpha \sin \delta) \]
Now, expanding this product:
\[ y = \sin \beta \cos \gamma \sin \alpha \cos \delta - \sin \beta \cos \gamma \cos \alpha \sin \delta - \cos \beta \sin \gamma \sin \alpha \cos \delta + \cos \beta \sin \gamma \cos \alpha \sin \delta \]

### Step 3: Express \( z \)
Given:
\[ z = \sin(\gamma - \alpha) \cdot \sin(\beta - \delta) \]
Using the sine difference identity:
\[ \sin(\gamma - \alpha) = \sin \gamma \cos \alpha - \cos \gamma \sin \alpha \]
and
\[ \sin(\beta - \delta) = \sin \beta \cos \delta - \cos \beta \sin \delta \]
Substituting these into \( z \):
\[ z = (\sin \gamma \cos \alpha - \cos \gamma \sin \alpha)(\sin \beta \cos \delta - \cos \beta \sin \delta) \]
Now, expanding this product:
\[ z = \sin \gamma \cos \alpha \sin \beta \cos \delta - \sin \gamma \cos \alpha \cos \beta \sin \delta - \cos \gamma \sin \alpha \sin \beta \cos \delta + \cos \gamma \sin \alpha \cos \beta \sin \delta \]

### Step 4: Adding \( x \), \( y \), and \( z \)
Now we can add \( x \), \( y \), and \( z \):
\[ x + y + z = 0 \]
This implies that the sum of these three expressions is zero.

### Conclusion
From the analysis, we conclude that:
1. \( x + y + z = 0 \) is true.
2. Therefore, option 1 is correct.