## Task A-3.1.

Solve the equation

$$
\cos (2 x)+\cos (2 y)+2 \sin x \sin y+2 \cos x \cos y=4
$$
## First Solution.

By applying the addition formula, we obtain the equivalent equation

$$
\cos (2 x)+\cos (2 y)+2 \cos (x-y)=4
$$

Since the cosine is at most 1, we conclude that $\cos (2 x)=\cos (2 y)=\cos (x-y)=1$. 4 points

It follows that $2 x=2 n \pi$ for some integer $n$ and $2 y=2 m \pi$ for some integer $m$.

Furthermore, $x-y=(n-m) \pi$ must be an integer multiple of $2 \pi$, so $n$ and $m$ must have the same parity.

Therefore, the solutions are $(x, y)=(2 k \pi, 2 l \pi)$ and $(x, y)=((2 k+1) \pi,(2 l+1) \pi)$ for $k, l \in \mathbb{Z}$.