Problem: Prove that for all real numbers x, y we have

Solution: Consider the function f (t) = arctan(t). Then the inequality we have to prove can be rewritten like this:

Thus we need to find out how large the fraction is, but that expression is exactly like something from the Mean value theorem. Are its assumptions satisfied? If we take any two numbers x < y, then the function arctangent is continuous on the interval [x,y] and differentiable on the interval (x,y). Thus MVT can be applied and it tells us that there is some c between x and y such that

Thus we can estimate

This proves the claim.

Remark: If we put y = 0 into the estimate we proved, we get the inequality

This is nothing surprising for x large, since we know that arctangent is bounded and so x must sooner or later outgrow it, but it is interesting that this estimate also works for small x. Similar estimate can be shown (using analogous procedure) for the function sine.

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