$\log \tan 1^0+\log \tan 2^0 + ...........+ \log \tan {89}^0 =$

$\begin{array}{rcl} & & \log \tan 1^0+\log \tan 2^0 + ...........+ \log \tan {89}^0 \\ & =& \log \tan 1° \tan 89° \tan 2° \tan 88°.....\tan 45°\\ & =& \log 1=0 (\because A+B={90}^o\Rightarrow \tan A \tan B=1)\end{array}$