Find all the pairs of x,y that satisfy the equation `cosx+cosy+cos(x+y)=-3/2`

Answer

To solve the equation \( \cos x + \cos y + \cos(x+y) = -\frac{3}{2} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the given equation:
\[ \cos x + \cos y + \cos(x+y) = -\frac{3}{2} \]

### Step 2: Use the cosine addition formula
We can use the cosine addition formula:
\[ \cos(x+y) = \cos x \cos y - \sin x \sin y \]
However, a more useful approach is to express \( \cos x + \cos y \) using the sum-to-product identities:
\[ \cos x + \cos y = 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) \]
Thus, we rewrite the equation:
\[ 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) + \cos(x+y) = -\frac{3}{2} \]

### Step 3: Substitute \( \cos(x+y) \)
Using the identity for \( \cos(x+y) \):
\[ \cos(x+y) = 2 \cos^2\left(\frac{x+y}{2}\right) - 1 \]
Substituting this back into the equation gives:
\[ 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) + 2 \cos^2\left(\frac{x+y}{2}\right) - 1 = -\frac{3}{2} \]

### Step 4: Simplify the equation
Rearranging the equation results in:
\[ 2 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) + 2 \cos^2\left(\frac{x+y}{2}\right) = -\frac{1}{2} \]
Multiplying through by 2 to eliminate the fraction:
\[ 4 \cos\left(\frac{x+y}{2}\right) \cos\left(\frac{x-y}{2}\right) + 4 \cos^2\left(\frac{x+y}{2}\right) = -1 \]

### Step 5: Set up a quadratic equation
Let \( z = \cos\left(\frac{x+y}{2}\right) \) and \( w = \cos\left(\frac{x-y}{2}\right) \):
\[ 4zw + 4z^2 + 1 = 0 \]
This can be rearranged as:
\[ 4z^2 + 4zw + 1 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 4, b = 4w, c = 1 \):
\[ z = \frac{-4w \pm \sqrt{(4w)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} \]
\[ z = \frac{-4w \pm \sqrt{16w^2 - 16}}{8} \]
\[ z = \frac{-4w \pm 4\sqrt{w^2 - 1}}{8} \]
\[ z = \frac{-w \pm \sqrt{w^2 - 1}}{2} \]

### Step 7: Find values of \( x \) and \( y \)
From \( z = \cos\left(\frac{x+y}{2}\right) \), we can find \( x+y \) and \( x-y \) using the values of \( z \) and \( w \).

### Step 8: General solutions
The general solutions for \( x \) and \( y \) can be expressed as:
\[ x = 2m\pi + \frac{\pi}{3} + n\pi \quad \text{and} \quad y = 2m\pi - \frac{\pi}{3} - n\pi \]
for integers \( m \) and \( n \).

### Final Answer
Thus, the pairs \( (x, y) \) that satisfy the equation are:
\[ (x, y) = \left(2m\pi + \frac{\pi}{3} + n\pi, 2m\pi - \frac{\pi}{3} - n\pi\right) \]
for integers \( m \) and \( n \).