`cos alphasin(beta-gamma)+cosbetasin(gamma-alpha)+cos gammasin(alpha-beta)=`

Options

A. 0
B. `1//2`
C. 1
D. `4cos alpha cos beta cos gamma`

Answer

To solve the expression \( \cos \alpha \sin(\beta - \gamma) + \cos \beta \sin(\gamma - \alpha) + \cos \gamma \sin(\alpha - \beta) \), we will use the sine difference identity and simplify step by step.

### Step 1: Apply the Sine Difference Identity
Using the identity \( \sin(a - b) = \sin a \cos b - \cos a \sin b \), we can rewrite each sine term in the expression.

\[ \sin(\beta - \gamma) = \sin \beta \cos \gamma - \cos \beta \sin \gamma \]
\[ \sin(\gamma - \alpha) = \sin \gamma \cos \alpha - \cos \gamma \sin \alpha \]
\[ \sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta \]

### Step 2: Substitute the Sine Terms Back into the Expression
Now we substitute these identities back into the original expression:

\[ \cos \alpha (\sin \beta \cos \gamma - \cos \beta \sin \gamma) + \cos \beta (\sin \gamma \cos \alpha - \cos \gamma \sin \alpha) + \cos \gamma (\sin \alpha \cos \beta - \cos \alpha \sin \beta) \]

### Step 3: Distribute the Cosine Terms
Distributing the cosine terms gives us:

\[ \cos \alpha \sin \beta \cos \gamma - \cos \alpha \cos \beta \sin \gamma + \cos \beta \sin \gamma \cos \alpha - \cos \beta \cos \gamma \sin \alpha + \cos \gamma \sin \alpha \cos \beta - \cos \gamma \cos \alpha \sin \beta \]

### Step 4: Combine Like Terms
Now we can group and combine like terms:

1. The terms involving \( \cos \alpha \sin \beta \cos \gamma \) and \( -\cos \gamma \cos \alpha \sin \beta \) cancel each other out.
2. The terms involving \( \cos \beta \sin \gamma \cos \alpha \) and \( -\cos \beta \cos \gamma \sin \alpha \) also cancel each other out.
3. The terms involving \( \cos \gamma \sin \alpha \cos \beta \) and \( -\cos \alpha \cos \beta \sin \gamma \) also cancel each other out.

After cancellation, we find that all terms cancel out, leading to:

\[ 0 \]

### Final Answer
Thus, the final result is:

\[ \cos \alpha \sin(\beta - \gamma) + \cos \beta \sin(\gamma - \alpha) + \cos \gamma \sin(\alpha - \beta) = 0 \]