Problems

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 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.

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?