If $\tan \theta +cot \theta =2, \mathrm{then} \mathrm{the} \mathrm{value} \mathrm{of} {\tan }^2\theta +co{t}^2\theta$is

$\mathrm{tan\theta }+\mathrm{cot\theta }=2$

Squaring both the sides

${\tan }^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }+2 \mathrm{tan\theta } \cot \mathrm{\theta } = 2^2$

${\tan }^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }+2 \mathrm{tan\theta }\times \frac{1}{\mathrm{tan\theta }}=4$

${\tan }^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }+2\left(1\right)=4$

${\tan }^2\theta +co{t}^2\theta +2=4$

${\tan }^2\theta +co{t}^2\theta =4-2$

${\tan }^2\mathrm{\theta }+{\cot }^2\mathrm{\theta }=2$

Hence, the correct answer is 2.