Which is greater, sin(cos(x)) or cos(sin(x))? Prove it!

Since the category is "Calculus", no doubt there must be some way calculus can help prove the cos(sin(x)) is always greater than sin(cos(x)).

Let f(x) = cos(sin(x)) - sin(cos(x))
We suspect f(x) is always positive.  Let's find the minimum value of f(x).
Start by finding the first derivative, f'(x), and set it equal to zero.

f'(x) = -sin(sin(x)) * cos(x) + cos(cos(x)) * sin(x)
which I haven't been able to break down further.
But from graphing it out, f'(x) is equal to zero at a series of well defined values for x:
0, pi, 2pi, ..., k*pi    and
L, 2pi + or - L, 4pi + or - L, .....
where L is approx:  .6927.... something
(and probably at negative values of x too, but I didn't check that)

So, we now have a limited number of x values to check to proove that cos(sin) is always greater than sin(cos).

Not totally rigorous, partly uses numeric methods.  It's enough to convince me.

Hopefully someone can provide a complete pure mathematical proof.

Posted by Larry on 2005-08-12 22:43:02