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?

The following is not completely rigourous, but can be made so with little effort.
An obvious trigonometric identity is $\sin^2 x+\cos^2 x=1$. Using this,
put $y=\sin^2 x$ to obtain a function of the form
$$f(y)=(y+\frac{1}{y})^2+(1-y+\frac{1}{1-y})^2$$
defined on the open interval $(0,1)$. This function is symmetric around $y=\frac{1}{2}$, because $f(y)=f(1-y)$. Also, $\lim_{y\to 0^+}f(y)=\lim_{y\to 1^-}f(y)=+\infty$. You can use a bit of calculus to convince yourself that $f$ decreases in $(0,1/2)$ and increases in $(1/2,1)$ to conclude that there is a minimum (unique, actually), when $y=1/2$.
