If ` 1 + cos x cos y + sin x sin y = 0`, then which of the following is / are true
1. `cosx + cosy = 0`
2. `sinx + siny = 0`
3. `sinx + cosy = 0`
chose the correct option from the code given below

Options

A. only 1
B. only 2
C. only 3
D. 1 and 2

Answer

To solve the equation \(1 + \cos x \cos y + \sin x \sin y = 0\), we can start by simplifying it.

### Step 1: Rewrite the equation
We can rewrite the given equation as:
\[ \cos x \cos y + \sin x \sin y = -1 \]
This can be recognized as the cosine of the sum of angles:
\[ \cos(x - y) = -1 \]

### Step 2: Analyze the cosine function
The cosine function equals \(-1\) at odd multiples of \(\pi\):
\[ x - y = (2k + 1)\pi \quad \text{for } k \in \mathbb{Z} \]
This implies:
\[ x = y + (2k + 1)\pi \]

### Step 3: Check the options
Now we need to check which of the given options are true based on our findings.

**Option 1: \( \cos x + \cos y = 0 \)**

Using the cosine addition formula:
\[ \cos x + \cos y = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]
Since \(x - y = (2k + 1)\pi\), we have:
\[ \frac{x - y}{2} = \frac{(2k + 1)\pi}{2} \quad \Rightarrow \quad \cos\left(\frac{x - y}{2}\right) = 0 \]
Thus, \( \cos x + \cos y = 0 \) is true.

**Option 2: \( \sin x + \sin y = 0 \)**

Using the sine addition formula:
\[ \sin x + \sin y = 2 \sin\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]
Again, since \(x - y = (2k + 1)\pi\):
\[ \cos\left(\frac{x - y}{2}\right) = 0 \]
Thus, \( \sin x + \sin y = 0 \) is also true.

**Option 3: \( \sin x + \cos y = 0 \)**

This option does not have a standard trigonometric identity that simplifies it directly. We cannot derive a conclusion about this expression from the previous findings.

### Conclusion
From the analysis:
- Option 1 is true: \( \cos x + \cos y = 0 \)
- Option 2 is true: \( \sin x + \sin y = 0 \)
- Option 3 is false: \( \sin x + \cos y = 0 \)

Thus, the correct options are **1 and 2**.

### Final Answer
The correct answer is **Option 4: 1 and 2**.

---