SCHWEITZER 1981 Problem 6

Let $f$ be a strictly increasing, continuous function mapping $I=[0,1]$ onto itself. Prove that the following inequality holds for all pairs $x,y \in I$ : $$1-\cos (xy) \leq \int_0^xf(t) \sin (tf(t))dt + \int_0^y f^{-1}(t) \sin (tf^{-1}(t)) dt .$$

Zs. Pales