Prove that if you rotate through an angle of \(\alpha\) with the center at the origin, the point with the coordinates \((x, y)\), it goes to the point \((x \cos \alpha - y \sin \alpha, x \sin \alpha + y \cos \alpha)\).
Does there exist a real number \({\alpha}\) such that the number \(\cos {\alpha}\) is irrational, and all the numbers \(\cos 2{\alpha}\), \(\cos 3{\alpha}\), \(\cos 4{\alpha}\), \(\cos 5{\alpha}\) are rational?