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 $|f(x+y)+\sin x+\sin y|<2$?

The quadratic trinomials $f(x)$ and $g(x)$ are such that $f^{\prime }(x)g^{\prime }(x)\ge |f(x)|+|g(x)|$ for all real $x$. Prove that the product $f(x)g(x)$ is equal to the square of some trinomial.