The solutions of the system of equations `sin x sin y=sqrt(3)/4, cos x cos y= sqrt(3)/4` are

Options

A. `x=pi/3+pi/2 (2n+k), n, k in I`
B. `y=pi/6+pi/2 (k-2n), n, k in I`
C. `x=pi/6+pi/2 (2n+k), n, k in I`
D. `y=pi/3+pi/2 (k-2n), n, k in I`

Explanation

To solve the system of equations given by:

1. \( \sin x \sin y = \frac{\sqrt{3}}{4} \)
2. \( \cos x \cos y = \frac{\sqrt{3}}{4} \)

we can follow these steps:

### Step 1: Add the two equations

We start by adding the two equations:

\[ \sin x \sin y + \cos x \cos y = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \]

### Step 2: Use the cosine addition formula

The left-hand side can be rewritten using the cosine addition formula:

\[ \sin x \sin y + \cos x \cos y = \cos(x - y) \]

Thus, we have:

\[ \cos(x - y) = \frac{\sqrt{3}}{2} \]

### Step 3: Solve for \(x - y\)

The general solutions for \( \cos \theta = \frac{\sqrt{3}}{2} \) are:

\[ x - y = 2n\pi \pm \frac{\pi}{6}, \quad n \in \mathbb{Z} \]

### Step 4: Subtract the two equations

Next, we subtract the first equation from the second:

\[ \cos x \cos y - \sin x \sin y = 0 \]

### Step 5: Use the cosine subtraction formula

This can be rewritten using the cosine subtraction formula:

\[ \cos(x + y) = 0 \]

### Step 6: Solve for \(x + y\)

The general solutions for \( \cos \theta = 0 \) are:

\[ x + y = \frac{\pi}{2} + k\pi, \quad k \in \mathbb{Z} \]

### Step 7: Solve the system of equations

Now we have two equations:

1. \( x - y = 2n\pi \pm \frac{\pi}{6} \)
2. \( x + y = \frac{\pi}{2} + k\pi \)

We can solve these equations simultaneously.

### Step 8: Express \(x\) and \(y\)

Adding the two equations:

\[ 2x = \left(2n\pi \pm \frac{\pi}{6}\right) + \left(\frac{\pi}{2} + k\pi\right) \]

This gives:

\[ x = n\pi + \frac{1}{2}\left(\frac{\pi}{2} + k\pi + 2n\pi \pm \frac{\pi}{6}\right) \]

Now, subtracting the two equations:

\[ 2y = \left(\frac{\pi}{2} + k\pi\right) - \left(2n\pi \pm \frac{\pi}{6}\right) \]

This gives:

\[ y = \frac{1}{2}\left(\frac{\pi}{2} + k\pi - 2n\pi \mp \frac{\pi}{6}\right) \]

### Final Solutions

Thus, the solutions for \(x\) and \(y\) can be summarized as:

\[ x = n\pi + \frac{\pi}{4} + \frac{\pi}{12} \quad \text{or} \quad x = n\pi + \frac{\pi}{4} - \frac{\pi}{12} \]

\[ y = \frac{\pi}{2} + k\pi - n\pi \mp \frac{\pi}{12} \]