The expression $\left(1+\tan x+{\tan }^{2}x\right)\left(1-\cot x+{\cot }^{2}x\right)$  has the positive values for $x$ , given by

$=\left(1+\tan x+{\tan }^{2}x\right)\left(1-\frac{1}{\tan x}+\frac{1}{{\tan }^{2}x}\right)$

$=\left(1+\tan x+{\tan }^{2}x\right)\frac{\left(1+{\tan }^{2}x-\tan x\right)}{{\tan }^{2}x}$

$=\frac{\left({\sec }^{2}x+\tan x\right)\left({\sec }^{2}x-\tan x\right)}{{\tan }^{2}x}$

$=\frac{\left({\sec }^{4}x-{\tan }^{2}x\right)}{{\tan }^{2}x}$

$\Rightarrow \frac{{\sec }^{4}x}{{\tan }^{2}x}-1$

$=\frac{1}{{\cos }^{4}x.\frac{{\sin }^{2}x}{{\cos }^{2}x}}-1$

$=\frac{4}{{\sin }^{2}2x}-1$

It’s positive for all values of $x$