If tan θ + cot θ = 6, then the value of tan²θ + cot²θ is ____ .

The correct option is 34.

Let us find the value of ${\tan }^2\theta +{\cot }^2\theta$ step-by-step.

We are given the following equation:
$\tan \theta +\cot \theta =6$

To find the value of the sum of their squares, we can square both sides of the given equation:
${\left(\tan \theta +\cot \theta \right)}^2=6^2$

Expanding the left side using the algebraic identity ${\left(a+b\right)}^2=a^2+b^2+2ab$, we get:
${\tan }^2\theta +{\cot }^2\theta +2\tan \theta \cot \theta =36$

Recall the fundamental trigonometric identity relating tangent and cotangent:
$\cot \theta =\frac{1}{\tan \theta }$
which implies:
$\tan \theta ·\cot \theta =1$

Substitute this identity back into our expanded equation:
${\tan }^2\theta +{\cot }^2\theta +2\left(1\right)=36$
${\tan }^2\theta +{\cot }^2\theta +2=36$

Subtracting 2 from both sides of the equation gives:
${\tan }^2\theta +{\cot }^2\theta =36-2$
${\tan }^2\theta +{\cot }^2\theta =34$

Therefore, the value of ${\tan }^2\theta +{\cot }^2\theta$ is 34.