<s>[INST] Minimizing $\left ( \sin^2(x) + \frac{1}{\sin^2(x)} \right )^2 + \left ( \cos^2(x) + \frac{1}{\cos^2(x)} \right )^2$ While solving a problem I came across this task, minimizing 
\begin{align}
\left ( \sin^2(x) + \frac{1}{\sin^2(x)} \right )^2 + \left ( \cos^2(x) + \frac{1}{\cos^2(x)} \right )^2.
\end{align}
One can easily do it with calculus to show that the minimum value is $12.5$.
I tried to do it using trigonometric identities and fundamental inequalities (like AM-GM, Cauchy-Schwarz, etc.) but failed. Can someone help me to do it using trig identities and inequalities?
 [/INST] Here is a solution using AM-GM and the double-angle formulae


*

*$(\star):\sin^2 x = \frac{1-\cos 2x}{2}$, $\cos^2 x = \frac{1+\cos 2x}{2}$
\begin{eqnarray*}\left ( \sin^2 x + \frac{1}{\sin^2 x} \right )^2 + \left ( \cos^2x + \frac{1}{\cos^2x} \right )^2
 & \stackrel{AM-GM}{\geq} & 2\left ( \sin^2x + \frac{1}{\sin^2x} \right ) \left ( \cos^2x+ \frac{1}{\cos^2x} \right ) \\
 & = & 2\frac{(\sin^4 x + 1)(\cos^4 x + 1)}{\sin^2 x \cdot \cos^2 x}\\
 & \stackrel{AM-GM}{\geq} & 8\frac{(\sin^4 x + 1)(\cos^4 x + 1)}{(\sin^2 x + \cos^2 x)^2}\\
 & \stackrel{\star}{=} & 8\left(\frac{(1-\cos 2x)^2}{4}+1 \right)\left(\frac{(1+\cos 2x)^2}{4}+1 \right)\\
 & = & \frac{1}{2}(5+\cos^2 2x - 2\cos 2x)(5+\cos^2 2x + 2\cos 2x)\\
 & = & \frac{1}{2}(25+\cos^4 2x +6 \cos^2 2x)\\
 & \geq & \frac{25}{2}
\end{eqnarray*}
Note that equality is attained for $\cos 2x = 0 \Leftrightarrow \cos^2 x = \sin ^2 x$, where the last condition is required to produce equality for AM-GM.
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