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.

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.
