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Find (x, y)

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anemone

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Feb 14, 2012
3,756
Find all pairs $(x, y)$ of real numbers with $0<x<\dfrac{\pi}{2}$ such that \(\displaystyle \frac{(\sin x)^{2y}}{(\cos x)^{\frac{y^2}{2}}}+\frac{(\cos x)^{2y}}{(\sin x)^{\frac{y^2}{2}}}=\sin (2x)\).
 
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anemone

MHB POTW Director
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Feb 14, 2012
3,756
Here is a solution that I found somewhere online:

By applying AM-GM inequality to the expression on the left, we obtain

\(\displaystyle \frac{(\sin x)^{2y}}{(\cos x)^{\frac{y^2}{2}}}+\frac{(\cos x)^{2y}}{(\sin x)^{\frac{y^2}{2}}} \ge 2 \sqrt{\frac{(\sin x \cos x)^{2y}}{(\sin x\cos x)^{\frac{y^2}{2}}}} \ge 2(\sin x \cos x)^{y-\frac{y^2}{4}} \)

and since \(\displaystyle \frac{(\sin x)^{2y}}{(\cos x)^{\frac{y^2}{2}}}+\frac{(\cos x)^{2y}}{(\sin x)^{\frac{y^2}{2}}}=\sin (2x)\)

\(\displaystyle \sin (2x) \ge 2(\sin x \cos x)^{y-\frac{y^2}{4}}\)

\(\displaystyle 2(\sin x \cos x) \ge 2(\sin x \cos x)^{y-\frac{y^2}{4}}\)

but \(\displaystyle \sin x \cos x=\frac{\sin 2x}{2} \le \frac{1}{2}\)

we must have \(\displaystyle y-\frac{y^2}{4}\ge 1\rightarrow (y-2)^2 \le 0\)

This is true iff $y=2$ and when $\sin x=\cos x$, so there is a unique solution to this problem where \(\displaystyle (x, y)=\left(\frac{\pi}{4}, 2 \right)\).