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cos(sin(x)) is always greater.
First, we only need to consider x in [-π,π] by periodicity. Second, because sin(cos(x)) and cos(sin(x)) are both even, we only need to consider x in [0,π].
If x is in [π/2,π], then sin(cos(x))≤ sin(0)=0 ≤cos(1) ≤cos(sin(x)).
So now suppose 0<x<π/2, and suppose sin(cos(x))=cos(sin(x)).
Let y=sin(x), z=cos(x), so that
0
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