To solve the equation \(4 \cos(x) \cos(y) = 3\), we need to isolate one of the variables. Here is the step-by-step process:

1. **Isolate one of the cosine terms**:  
   \[
   \cos(x) \cos(y) = \frac{3}{4}
   \]

2. **Use a trigonometric identity**:  
   Recall that \(\cos(A) \cos(B) = \frac{1}{2} [\cos(A+B) + \cos(A-B)]\). Applying this identity:
   \[
   \frac{1}{2} [\cos(x+y) + \cos(x-y)] = \frac{3}{4}
   \]

3. **Multiply both sides by 2 to simplify**:
   \[
   \cos(x+y) + \cos(x-y) = \frac{3}{2}
   \]

This equation suggests that the sum of the cosines of the sum and difference of \(x\) and \(y\) equals \(\frac{3}{2}\). 

To find specific solutions, let's consider particular values for \(x\) and \(y\) that satisfy the equation. 

One possible approach is to assume \(x = y\). This gives us:

4. **Set \(x = y\)**:
   \[
   \cos(2x) + \cos(0) = \frac{3}{2}
   \]
   Since \(\cos(0) = 1\):
   \[
   \cos(2x) + 1 = \frac{3}{2}
   \]
   \[
   \cos(2x) = \frac{1}{2}
   \]

5. **Solve for \(x\)**:
   \[
   2x = \pm \frac{\pi}{3} + 2k\pi \quad \text{for integer } k
   \]
   \[
   x = \pm \frac{\pi}{6} + k\pi \quad \text{for integer } k
   \]

Thus, one set of solutions is:
\[
x = \frac{\pi}{6} + k\pi \quad \text{and} \quad y = \frac{\pi}{6} + k\pi \quad \text{for integer } k
\]

There could be other sets of solutions if we do not assume \(x = y\). However, the above provides a specific solution pair for \(x\) and \(y\). 

Would you like a more detailed exploration of other potential solutions, or do you have any other questions?

Here are some related questions to further explore this topic:

1. What are the general solutions for \(\cos(A) = \frac{1}{2}\)?
2. How would you solve \(\cos(x+y) + \cos(x-y) = \frac{3}{2}\) for other specific values of \(x\) and \(y\)?
3. Can \(\cos(x) \cos(y) = \frac{3}{4}\) be solved using complex numbers?
4. What other trigonometric identities could simplify \(\cos(x) \cos(y) = \frac{3}{4}\)?
5. How would you graph the solution set of \(\cos(x) \cos(y) = \frac{3}{4}\)?
6. Are there any specific angles where \(\cos(x) = \frac{\sqrt{3}}{2}\)?
7. What is the significance of periodicity in trigonometric functions in finding solutions?
8. How do the solutions change if we change the equation to \(4 \cos(x) \cos(y) = k\), where \(k\) is a different constant?

**Tip:** When solving trigonometric equations, always consider using identities and periodic properties of the trigonometric functions to find all possible solutions.

Learn how to solve the trigonometric equation 4cos(x)cos(y) = 3 step-by-step with detailed explanations and solutions. This problem involves applying trigonometric identities and solving for specific values of x and y. Suitable for high school students studying trigonometry.

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