RO-NMO 2011 Problem 101

Let $f:\mathbb{R}\longrightarrow\mathbb{R}$ a function having the property that $$\left| f(x+y)+\sin x+\sin y \right|\le 2,$$ for all real numbers $x,y.$ a) Prove that $\left| f(x) \right|\le 1+\cos x,$ for all real numbers $x.$ b) Give an example of what $f$ may be, if the interval $\left( -\pi ,\pi \right)$ is included in its support.