\section*{Problem 3 - 031223}

Determine the set of all pairs \((x, y)\) of real numbers \(x, y\) that satisfy the following equations:

\[
\cos \frac{x+y}{2} \cdot \cos \frac{x-y}{2}=\frac{1}{2} \quad ; \quad \cos x \cdot \cos y=\frac{1}{4}
\]
}

With the help of the addition theorem

\[
\cos \frac{x+y}{2} \cdot \cos \frac{x-y}{2}=\frac{1}{2}(\cos x+\cos y)
\]

one obtains the following equivalent system of equations to the given system:

\[
\cos x+\cos y=1 \quad ; \quad \cos x \cos y=\frac{1}{4}
\]

From this it follows:

\[
(1-\cos y) \cdot \cos y=\frac{1}{4} \quad \text { and } \quad\left(\cos y-\frac{1}{2}\right)^{2}=0
\]

i.e., \(\cos x=\frac{1}{2}\) and \(\cos y=\frac{1}{2}\). This is the only solution to the system of equations (1). From this it follows:

\[
x=2 m \pi \pm \frac{\pi}{3} \quad ; \quad y=2 n \pi \pm \frac{\pi}{3}
\]

where \(m\) and \(n\) are integers.

Taken from \([2]\)