Problems

Does there exist a function $f(x)$ defined for all $x\in \mathbb{R}$ and for all $x,y\in \mathbb{R}$ satisfying the inequality $|f(x+y)+\sin x+\sin y|<2$?

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?

Calculate ${\int }_0^{\pi /2}({\sin }^2(\sin x)+{\cos }^2(\cos x))dx$.

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 irrationality of the following numbers:

a) $\sqrt{3}17$

b) $\sqrt{2}+\sqrt{3}$

c) $\sqrt{2}+\sqrt{3}+\sqrt{5}$

d) $\sqrt{3}3-\sqrt{2}$

e) $\cos {10}^{\circ }$

f) $\tan {10}^{\circ }$

g) $\sin 1^{\circ }$

h) ${\log }_{2}3$

Prove that for $x\ne \pi n$ ($n$ is an integer) $\sin x$ and $\cos x$ are rational if and only if the number $\tan x/2$ is rational.

A square grid on the plane and a triangle with vertices at the nodes of the grid are given. Prove that the tangent of any angle in the triangle is a rational number.

Find the largest and smallest values of the functions

a) $f_1(x)=a\cos x+b\sin x$; b) $f_2(x)=a{\cos }^{2}x+b\cos x\sin x+c{\sin }^{2}x$.

Prove that the function $\cos \sqrt{x}$ is not periodic.

Prove the formulae: $\arcsin (-x)=-\arcsin x$, $\arccos (-x)=\pi -\arccos x$.