$\mathrm{logtan}{1}^0+\mathrm{logtan}{2}^0+···+\mathrm{logtan}{89}^0=$

$\tan \theta =\cot (90-\theta )\text{ and }\tan \theta \cdot \cot \theta =1$

Thus, $\tan {89}^{\circ }=\cot \left(90-{89}^{\circ }\right)=\cot 1^{\circ }$

$\text{ similarly, }\tan {88}^{\circ }=\cot 2^{\circ },\tan {87}^{\circ }=\cot 3^{\circ }\text{ and so on ···}$

$$\begin{gathered} =\log \tan 1^{\circ }+\log \tan {89}^{\circ } \\ =\log \left(\tan 1^{\circ }\times \tan {89}^{\circ }\right) \end{gathered}$$

$=\log \left(\tan 1^{\circ }\times \cot 1^{\circ }\right)$

=0

Similarly

$$\begin{gathered} =\log \left(\tan 2^{\circ }\right)+\log \left(\tan {88}^{\circ }\right) \\ =\log \left(\tan 2^{\circ }\times \cot 2^{\circ }\right) \\ =\log 1 \\ =0 \end{gathered}$$

$\mathrm{logtan}{1}^0+\mathrm{logtan}{2}^0+···+\mathrm{logtan}{89}^0=0$