The value of
\(e^{\log _{10} \tan 1^{\circ}+\log _{10} \tan 2^{\circ}+\log _{10} \tan 3^{\circ}+\ldots+\log _{10} \tan 89^{\circ}}\)
is

We know that
\(\begin{aligned} & \tan \theta \tan \left(90^{\circ}-\theta\right)=\tan \theta \cdot \cot \theta=1 \\ & \tan 1^{\circ} \tan 89^{\circ}=\tan 2^{\circ} \tan 88^{\circ}=\tan 3^{\circ} \tan 87^{\circ}\end{aligned}\)
\(=\tan 44^{\circ} \tan 46^{\circ}=1\)
So, \(\log _{10} \tan 1^{\circ}+\log _{10} \tan 2^{\circ} \ldots \log _{10} \tan 89^{\circ}\)
\(\begin{aligned}=\left(\log _{10} \tan 1^{\circ}+\log _{10} \tan 89^{\circ}\right) & \\ +\left(\log _{10} \tan 2^{\circ}\right. & \left.+\log _{10} \tan 88^{\circ}\right) \\ +\ldots+\left(\log _{10} \tan 44^{\circ}\right. & \left.+\log _{10} \tan 46^{\circ}\right) \\ & +\log _{10} \tan 45^{\circ}\end{aligned}\)
\(\begin{array}{r}=\log _{10}\left(\tan 1^{\circ}+\tan 89^{\circ}\right)+\log _{10}\left(\tan 2^{\circ}+\tan 88^{\circ}\right) \\ +\ldots+\log _{10}\left(\tan 44^{\circ}+\tan 46^{\circ}\right) \\ +\log _{10} \tan 45^{\circ}=\log _{10} 1+\ldots+\log _{10} I+\log _{10} 1\end{array}\)
\(=0\)
Thus, \(e^{\left.e \log _{10} \tan 1^{\circ}+\log _{10} \tan 2^{\circ}+\ldots+\log _{10} \tan 89^{\circ}\right)}\)
\(\left[\because \log _{10} 1=0\right]\)
\(=e^0=1\)