$\text{Given that : $\tan\theta+\cot\theta=2$}$
$\because$ $\cot\theta=\frac{1}{\tan\theta}$
$\implies$ $\tan\theta+\frac{1}{\tan\theta}=2$
$\implies$ $\frac{tan^2\theta+1}{\tan\theta}=2$
$\implies$ $\tan^2\theta+1=2\tan\theta$
$\implies$ $(\tan\theta-1)^2=0$
$\implies$ $\tan\theta=1$
$\because \tan45^\circ=1$
$\therefore \tan\theta=\tan45^\circ$
$\implies$ $\theta=45^\circ$
$\therefore$ $\tan^7\theta+cot^7\theta=(\tan45^\circ)^7+(\cot45^\circ)^7=1+1=2$
$\because tan45^\circ=cot45^\circ=1$
$\therefore$ $\tan^7\theta+cot^7\theta=1+1=2$
$\text{Option C is correct.}$