Prove that the function \(\cos \sqrt {x}\) is not periodic.
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 \((x \cos \alpha - y \sin \alpha, x \sin \alpha + y \cos \alpha)\).
Prove the formulae: \(\arcsin (- x) = - \arcsin x\), \(\arccos (- x) = \pi - \arccos x\).
Find the locus of points whose coordinates \((x, y)\) satisfy the relation \(\sin(x + y) = 0\).
Does there exist a flat quadrilateral in which the tangents of all interior angles are equal?
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?