<s>[INST] Finding the range of $\frac{1+\sin^4 x}{\sin^4 x}\cdot \frac{1+\cos^4 x}{\cos^4 x}$ 
Range of function $$f(x) = \frac{(1+\sin^4 x)}{\sin^4 x}\cdot \frac{(1+\cos^4 x)}{\cos^4 x}$$ 

Using $\bf{A.\geq G.M}$ Inequality,
$$1+\sin^4 x\geq 2\sin^2 x \qquad\text{and}\qquad 1+\cos^4 x\geq 2\cos^2 x$$
So $$\frac{(1+\sin^4 x)}{\sin^4 x}\cdot \frac{(1+\cos^4 x)}{\cos^4 x}\geq \frac{4}{\sin^2 x\cdot \cos^2 x} = \frac{16}{(\sin 2x)^2}\geq 16$$
But the answer  given as $\left[25,\infty\right)$.
Please help me. How can I solve it using an inequality or any other way. Thanks.
 [/INST] Answering to the question of why the OP method gives a lower bound for the codomain, consider (e.g.) the following:
$$
\begin{gathered}
  \left( {z^{\,2}  + 1} \right) \geqslant 2z\quad \left( {y^{\,2}  + 1} \right) \geqslant 2y\quad  \Rightarrow \quad \left( {z^{\,2}  + 1} \right)\left( {y^{\,2}  + 1} \right) \geqslant 4\,z\,y\quad  \Rightarrow  \hfill \\
   \Rightarrow \quad \left( {z^{\,2}  + 1} \right)\left( {y^{\,2}  + 1} \right) = 4\quad \left| {\;\left| z \right| = 1\; \wedge \;\left| y \right| = 1} \right. \hfill \\ 
\end{gathered} 
$$
Here, since each single inequality bears the $ \geqslant$ sign, implies that the equality is attained for some values of the variable.
When multiplied together , keeping the variables distinct , then surely for all
the $(y,z)$ combining the above values, the equality will be attained.
However, when in it we place $(y(x), z(x))$ we are going to restrict the values of
$y$ and $z$ to stay on the curve parametrized in $x$, which might not allow
$y$ and $z$ to attain "contemporarily" the equality (allow me to say that you might have $( = )\, \times \, ( > )$).
In particular, the substitution 
$$
\left\{ \begin{gathered}
  y = 1/\sin ^{\,2} x \hfill \\
  z = 1/\cos ^{\,2} x \hfill \\ 
\end{gathered}  \right.
$$
will produce the situation which is well evidentiated in the picture.  

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