题目

求证：cos(sinx)大于sin(cosx)
谢谢大家了.这道题希望能解答的详细点.在这里感谢大家.

提问时间：2021-01-11

答案

老外的办法,你顺便学英语了
We have to prove :
cos (sinx) - sin (cosx) > 0
=> cos (sinx) - cos ( π/2 - cosx ) > 0
=> 2 sin [ (π/4) + (1/2).(sinx - cosx) ].sin [ (π/4) - (1/2).(sinx - cosx) ] > 0 .(1)
If we could prove that both the factors on the left hand side of (1) are positive then the result obtained above (1) is proved.
Since | sinx - cosx | = | √2 sin (x-π/4) | ≤ √2 < π/2
We have - π/2 < ( sinx - cosx ) < π/2
=> - π/4 < ( sinx - cosx )/2 < π/4
So that 0 < π/4 + (1/2) ( sinx - cosx ) < π/2
And therefore sin [ (π/4) + (1/2).(sinx - cosx) ].> 0 ie Positive.
Similarly we can prove that
sin [ (π/4) - (1/2).(sinx - cosx) ] > 0 ie Positive
Hence (1) is true..Proved.