<s>[INST] Find all $(x,y)$ satisfying $(\sin^2x+\frac{1}{\sin^2 x})^2+(\cos^2x+\frac{1}{\cos^2 x})^2=12+\frac{1}{2}\sin y$ Find all pairs $(x,y)$ of real numbers that satisfy the equation $(\sin^2x+\frac{1}{\sin^2 x})^2+(\cos^2x+\frac{1}{\cos^2 x})^2=12+\frac{1}{2}\sin y$

I supposed $a=\sin^2x$ and $b=\cos^2x$
So the equation becomes $(a+\frac{1}{a})^2+(b+\frac{1}{b})^2=12+\frac{1}{2}\sin y$
As $a+\frac{1}{a}\geq 2$ and $b+\frac{1}{b}\geq 2$
$12+\frac{1}{2}\sin y\geq 8$
$\sin y\geq -8$
I am stuck here.I could not solve further.Please help me.Thanks.
 [/INST] As JMoravitz notes in the comments, it's probably the case that the minimum value of the LHS is 12.5. Let's prove this.
Using the inequality $a^2+b^2\ge\frac{(a+b)^2}{2}$, note that
\begin{align*}\left(\sin^2x + \frac{1}{\sin^2x}\right)^2 + \left(\cos^2x + \frac{1}{\cos^2x}\right)^2&\ge \frac{1}{2}\left(\sin^2x + \frac{1}{\sin^2x} + \cos^2x + \frac{1}{\cos^2x}\right)^2\\
&=\frac{1}{2}\left(1 + \frac{\cos^2x+\sin^2x}{\sin^2x\cos^2x}\right)^2 \\
&=\frac{1}{2}\left(1 + \frac{4}{(2\sin x\cos x)^2}\right)^2 \\
&=\frac{1}{2}\left(1 + \frac{4}{\sin^2{2x}}\right)^2\\
&\ge\frac{1}{2}\left(1 + \frac{4}{1}\right)^2 = \frac{25}{2}.
\end{align*}
I leave it to you to determine when this equality holds.
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