Solution :
Given: $tan \theta+cot \theta =2 $
$\tan \theta + \large\frac{1}{\tan \theta }$$=2$
$\tan ^2 \theta+1= 2 \tan \theta$
$\tan ^2 \theta -2 \tan \theta +1=0$
$\tan \theta -1=0$
$\tan \theta =1$
and $\cot \theta = \large\frac{1}{\tan \theta } $$=\frac{1}{1}$
$\tan ^{25} \theta+ \cot ^{25} \theta = (\tan \theta)^{25} +(\cot \theta )^{25} $
$\qquad= (1)^{25} +(1)^{25} $$=1+1=2$