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If 0 ≤ θ ≤ π/2, Prove the inequality cos(sinθ) > sin (cosθ).

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If 0 ≤ θ ≤ π/2, Prove the inequality cos(sinθ) > sin (cosθ).

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We have sinθ + cosθ = √2 sin(θ + π/4) ≤ √2 since the maximum value of sin(θ + π/4) = 1

But √2  < π/2; (√2 is approximately 1.414 and π/2 is approximately 1.59). Therefore,

cosine being a decreasing function in first quadrant. That is 

cos(sinθ) > sin(cosθ)

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