If `sin x+siny+sinz=3`. Find the value of `cos x+cosy+cosz`.

Answer

To solve the problem, we start with the given equation:

\[ \sin x + \sin y + \sin z = 3 \]

### Step 1: Analyze the Sine Values
We know that the sine function has a maximum value of 1. Therefore, for the equation \(\sin x + \sin y + \sin z = 3\) to hold true, each sine term must equal its maximum value:

\[ \sin x = 1, \quad \sin y = 1, \quad \sin z = 1 \]

### Step 2: Determine the Angles
The sine function equals 1 at specific angles. The primary angle where \(\sin\) equals 1 is:

\[ x = \frac{\pi}{2} + 2k\pi, \quad y = \frac{\pi}{2} + 2m\pi, \quad z = \frac{\pi}{2} + 2n\pi \]

for integers \(k\), \(m\), and \(n\).

### Step 3: Calculate the Cosine Values
Now we need to find the cosine values corresponding to these angles. We know that:

\[ \cos\left(\frac{\pi}{2}\right) = 0 \]

Thus, we have:

\[ \cos x = 0, \quad \cos y = 0, \quad \cos z = 0 \]

### Step 4: Sum the Cosine Values
Now, we can sum these cosine values:

\[ \cos x + \cos y + \cos z = 0 + 0 + 0 = 0 \]

### Conclusion
Therefore, the value of \(\cos x + \cos y + \cos z\) is:

\[ \boxed{0} \]

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