If `sin x+siny+sinz=0=cosx+cosy+cosz`, then find the value of `cos(theta-x)+cos(theta-y)+cos(theta-z)`

Explanation

To solve the problem, we need to find the value of \( \cos(\theta - x) + \cos(\theta - y) + \cos(\theta - z) \) given that \( \sin x + \sin y + \sin z = 0 \) and \( \cos x + \cos y + \cos z = 0 \).

### Step-by-Step Solution:

1. **Write down the given equations:**
\[ \sin x + \sin y + \sin z = 0 \]
\[ \cos x + \cos y + \cos z = 0 \]

2. **Use the cosine subtraction formula:**
The cosine subtraction formula states:
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
Therefore, we can expand each term:
\[ \cos(\theta - x) = \cos \theta \cos x + \sin \theta \sin x \]
\[ \cos(\theta - y) = \cos \theta \cos y + \sin \theta \sin y \]
\[ \cos(\theta - z) = \cos \theta \cos z + \sin \theta \sin z \]

3. **Combine the expanded terms:**
Adding these three equations together:
\[ \cos(\theta - x) + \cos(\theta - y) + \cos(\theta - z) = (\cos \theta \cos x + \sin \theta \sin x) + (\cos \theta \cos y + \sin \theta \sin y) + (\cos \theta \cos z + \sin \theta \sin z) \]

4. **Group the terms:**
Group the cosine and sine terms:
\[ = \cos \theta (\cos x + \cos y + \cos z) + \sin \theta (\sin x + \sin y + \sin z) \]

5. **Substitute the given values:**
From the problem, we know:
\[ \cos x + \cos y + \cos z = 0 \]
\[ \sin x + \sin y + \sin z = 0 \]
Therefore, substituting these values:
\[ = \cos \theta \cdot 0 + \sin \theta \cdot 0 \]
\[ = 0 + 0 = 0 \]

### Final Answer:
Thus, the value of \( \cos(\theta - x) + \cos(\theta - y) + \cos(\theta - z) \) is:
\[ \boxed{0} \]