Boundedness of approximate trigonometric functions

In this work, we prove that approximate trigonometric functions are bounded. That is, if a non-zero function ff satisfies the inequality |f(x+y)−f(x−y)−2f(x)f(y)|≤φ(x)|f(x+y)−f(x−y)−2f(x)f(y)|≤φ(x) or φ(y)φ(y), then ff is bounded.