Given  $f\left(\tan 2x\right)={\tan }^{4}x+\frac{1}{{\tan }^{4}x}\forall x\in (0,\frac{\pi }{4})$ and if the value of $f\left(\sin x\right)+f\left(\cos x\right)whenx=\frac{\pi }{4}$   is  $k$  then which is/are correct

Let  $t=\tan 2x=-\frac{2}{\tan x-\frac{1}{\tan x}},t>0$
 $\therefore \tan x-\frac{1}{\tan x}=\frac{-2}{t}$
 ${\tan }^{4}x+\frac{1}{{\tan }^{4}x}={\{{(\tan x-\frac{1}{\tan x})}^2+2\}}^2-2$
 $\therefore f\left(t\right)={(\frac{4}{t^2}+2)}^2-2$
$f\left(\sin x\right)+f\left(\cos x\right)={(\frac{4}{{\sin }^{2}x}+2)}^2+{(\frac{4}{{\cos }^{2}x}+2)}^2-4$ 
$\ge {(8+2)}^2+{(8+2)}^2-4\ge 196$              (At $x=\frac{\pi }{4}$   min occurs)

$\therefore k=196$