First, note that since sin(x) is between -1 and 1, and cos(y) is positive for y between -1 and 1, so cos(sin(x)) is positive for all x.

Next, consider the following:

D = {cos (sin x)}² - {sin (cos x)}²

= 1 - {sin (sin x)}² - {sin (cos x)}².

Note that for all x, (sin x)² <= [less than or equal to] x², so

D >= 1 - (sin x)² - (cos x)² = 0.

Thus, since cos(sin x) > 0, this means

cos(sin x) >= sin(cos x) for all x.

Also, it follows that (sin x)² = x² only for x = 0, and

cos(sin 0) = 1 > sin(cos 0) ~ .8415

Therefore, cos(sin x) > sin(cos x) for all real x.