This article is from the Puzzles FAQ, by Chris Cole email@example.com and Matthew Daly firstname.lastname@example.org with numerous contributions by others.
Show that (sin x)^(sin x) < (cos x)^(cos x) when 0 < x < pi/4.
The function f(x) = x^(1/sqrt(1-x^2)) is monotonically increasing for
0 < x < 1, easily verified by taking the derivative.
Since 0 < sin x < cos x < 1 for 0 < x < pi/4, f(sin x) < f(cos x).
But f(sin x) = (sin x)^(1/cos x) and f(cos x) = (cos x)^(1/sin x).
Raising both sides to the power (cos x.sin x), we get the desired