cos(cos x) - sin(sin x)
= cos(cos x) - cos( π/2 - sin x)
= - 2 sin[ π/4 + (cosx-sinx)/2] sin[ (cosx+sinx)/2 - π/4]
= 2 sin[ π/4 + (cosx-sinx)/2] sin[ π/4 - (cosx+sinx)/2 ]
| (cosx-sinx)/2 | ≤ √2 /2 < π/4 ,| (cosx+sinx)/2 | ≤ √2 /2 < π/4
=> sin[ π/4 + (cosx-sinx)/2] >0 且 sin[ π/4 - (cosx+sinx)/2 ] >0
=> cos(cos x) - sin(sin x) > 0
即证.