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I am new here ,can anybody help to solve this problem: How to compare $\left(\sin \left(x\right)\right)^{\cos \left(x\right)}$ and $ \left(\cos \left(x\right)\right)^{\sin \left(x\right)}$ in the interval $\left[ 0,\frac { \pi }{ 2 } \right] $

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  • $\begingroup$ Alternatively, you can use the proper definition of exponentiation and write: $$e^{\cos x \ln \sin x},~~~~~e^{\sin x \ln \cos x}$$ $\endgroup$ – Yuriy S Jul 6 '16 at 12:40
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Maybe a plot helps (blue is $\sin^{\cos(x)}(x)$):

enter image description here

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  • $\begingroup$ @feynman You're welcome, I'm glad that I could help! $\endgroup$ – Jan Jul 6 '16 at 12:23
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Equivalently we may compare both quantities raised to the same positive number, $\dfrac1{\sin x \cos x}$.

So now we need to compare $(\sin x)^{\frac1{\sin x}}$ with $(\cos x )^{\frac1{\cos x}}$. It is now enough to note that the function $t^{1/t}$ is increasing in $(0,1)$, so this is equivalent to comparing $\sin x$ and $\cos x$ in the interval $[0, \frac\pi 2]$, which is obvious.

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They only have the same value at $\frac {\pi }{ 4 }$ in the given interval. Now we check $(\cos (0))^{\sin(0) }= 1,( \sin (0))^{\cos (0) }=0$ hence $(\cos (x))^{\sin (x)} $ above the other in $[0, \pi/4] $. Check their value at $\pi/2$.

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