If ${\tan }^4\theta +{\tan }^2\theta =1$, what is the value of $11({\cos }^4\theta +{\cos }^2\theta )$?

To find the value of the expression $11({\cos }^4\theta +{\cos }^2\theta )$, we start with the given trigonometric equation:
${\tan }^4\theta +{\tan }^2\theta =1$

We can factor out ${\tan }^2\theta$ from the left side of the equation:
${\tan }^2\theta ({\tan }^2\theta +1)=1$

Using the fundamental trigonometric identity $1+{\tan }^2\theta ={\sec }^2\theta$, we can substitute this into the equation:
${\tan }^2\theta \cdot {\sec }^2\theta =1$

Next, we express $\tan \theta$ and $\sec \theta$ in terms of $\sin \theta$ and $\cos \theta$:
$\tan \theta =\frac{\sin \theta }{\cos \theta }$ and $\sec \theta =\frac{1}{\cos \theta }$

Substituting these definitions, the equation becomes:
$\frac{{\sin }^2\theta }{{\cos }^2\theta }\cdot \frac{1}{{\cos }^2\theta }=1$
$\frac{{\sin }^2\theta }{{\cos }^4\theta }=1$
Multiplying both sides by ${\cos }^4\theta$, we obtain:
${\sin }^2\theta ={\cos }^4\theta$

Now, we substitute this relation into the expression we want to evaluate: ${\cos }^4\theta +{\cos }^2\theta$.
Replacing ${\cos }^4\theta$ with ${\sin }^2\theta$:
${\cos }^4\theta +{\cos }^2\theta ={\sin }^2\theta +{\cos }^2\theta$

By the standard Pythagorean trigonometric identity, we know that:
${\sin }^2\theta +{\cos }^2\theta =1$
Therefore, we have:
${\cos }^4\theta +{\cos }^2\theta =1$

Finally, we multiply this value by 11:
$11({\cos }^4\theta +{\cos }^2\theta )=11\left(1\right)=11$

Thus, the correct option is 11.