Inequalities

Can you prove our inequality holds for all values of x and y between 0 and 1?

Can you solve this inequalities challenge?

Farey sequences are lists of fractions in ascending order of magnitude. Can you prove that in every Farey sequence there is a special relationship between Farey neighbours?

Which is larger cos(sin x) or sin(cos x) ? Does this depend on x ?

Here the diagram says it all. Can you find the diagram?