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Question

For all real θ, prove cos(sin θ) > sin(cos θ)

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Solution

We have to prove
cos(sinθ)sin(cosθ)>0
or cos(sinθ)cos(π2cosθ)>0
or 2sin(π4+sinθcosθ2)
sin(π4+sinθcosθ2)>0....(1)
We now show that both the factors on the left hand side of (1) are positive. since,
|sinθcosθ|=2sin(θπ4)2π2
We have π2<sinθcosθ<π2
π4<sinθcosθ2<π4
Add π4 so that 0<π4<sinθcosθ2<π2
and therefore sin(π4+sinθcosθ2)>0
Similarly we can prove sin(π4sinθ+cosθ2)>0
Hence (1) holds which is what we wanted to prove.

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