Prove that for all real and
Notice first that . So by adding multiples of pi to x and y, we may assume that both x and y lie between 0 and pi. Also, if then we can replace x by and y by . That will have the effect of replacing x+y by , and again the values of , and will be unchanged. Therefore we can assume that and hence x, y and are the three angles of a triangle.
So we want to prove that , where A, B, C are the angles of a triangle with sides a, b, c. In fact, the stronger inequality is true.
The cosine rule says that
So we want to prove that . But we know that , and and therefore . If you multiply that out, you will find that it gives exactly the inequality that is wanted.