comparing the smallest positive roots of $\cos (\sin x) = x$ and $\sin (\cos x) = x$

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comparing the smallest positive roots of   cos  ⁡  (  sin  ⁡  x  )  =  x and   sin  ⁡  (  cos  ⁡  x  )  =  x

wayembesail · Accepted Answer

Let&#039;s argue that   cos  ⁡  (  sin  ⁡  x  )  &amp;gt;  sin  ⁡  (  cos  ⁡  x  ) at least on   [  0  ,      π    2    ].Suppose   x  ∈  [  0  ,      π    2  First, we know   sin  ⁡  x  ≤  x so   sin  ⁡  (  cos  ⁡  x  )  ≤  cos  ⁡  x with equality only when   cos  ⁡  x  =  0, i.e., at   x  =      π    2  .Second, since   cos  ⁡  x is decreasing, and   sin  ⁡  x  &amp;lt;=  x,   cos  ⁡  (  sin  ⁡  x  )  ≥  cos  ⁡  x, with equality only then   sin  ⁡  x  =  0, i.e., at x=0.Thus,   cos  ⁡  (  sin  ⁡  x  )  &amp;gt;  sin  ⁡  (  cos  ⁡  x  )  .Hence,   y  =  cos  ⁡  (  sin  ⁡  x  ) intersects y=x at a larger value of x than   y  =  sin  ⁡  (  cos  ⁡  x  ) does.

comparing the smallest positive roots of cos ⁡ ( sin ⁡<!-- ⁡

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