Jorden Pace

2022-07-03

Find all (x,y) satisfying $\left({\mathrm{sin}}^{2}x+\frac{1}{{\mathrm{sin}}^{2}x}{\right)}^{2}+\left({\mathrm{cos}}^{2}x+\frac{1}{{\mathrm{cos}}^{2}x}{\right)}^{2}=12+\frac{1}{2}\mathrm{sin}y$

Jamarcus Shields

Using the inequality ${a}^{2}+{b}^{2}\ge \frac{\left(a+b{\right)}^{2}}{2}$, note that
$\begin{array}{rl}{\left({\mathrm{sin}}^{2}x+\frac{1}{{\mathrm{sin}}^{2}x}\right)}^{2}+{\left({\mathrm{cos}}^{2}x+\frac{1}{{\mathrm{cos}}^{2}x}\right)}^{2}& \ge \frac{1}{2}{\left({\mathrm{sin}}^{2}x+\frac{1}{{\mathrm{sin}}^{2}x}+{\mathrm{cos}}^{2}x+\frac{1}{{\mathrm{cos}}^{2}x}\right)}^{2}\\ & =\frac{1}{2}{\left(1+\frac{{\mathrm{cos}}^{2}x+{\mathrm{sin}}^{2}x}{{\mathrm{sin}}^{2}x{\mathrm{cos}}^{2}x}\right)}^{2}\\ & =\frac{1}{2}{\left(1+\frac{4}{\left(2\mathrm{sin}x\mathrm{cos}x{\right)}^{2}}\right)}^{2}\\ & =\frac{1}{2}{\left(1+\frac{4}{{\mathrm{sin}}^{2}2x}\right)}^{2}\\ & \ge \frac{1}{2}{\left(1+\frac{4}{1}\right)}^{2}=\frac{25}{2}.\end{array}$

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