Aristarchus's inequality

Aristarchus's inequality is a law of trigonometry which states that if α and β are acute angles and β < α then
Ptolemy used the first of these inequalities while constructing his table of chords.

Proof

The proof is a consequence of the more widely known inequalities

Proof of the first inequality

Using these inequalities we can first prove that
We first note that the inequality is equivalent to
which itself can be rewritten as
We now want to show that
The second inequality is simply. The first one is true because

Proof of the second inequality

Now we want to show the second inequality, i.e. that:
We first note that due to the initial inequalities we have that:
Consequently, using that in the previous equation we obtain:
We conclude that