Question

The value of $\sum _{r=1}^{89}{\log }_3\left(\tan \left(r^o\right)\right)$ is

• A. $\log \sqrt{3}$
• B. $\log \sqrt{2}$
• C. $1$
• D. $0$

Answer 1

The correct option is D $0$

Let $S=\sum _{r=1}^{89}{\log }_3\left(\tan \left(r^o\right)\right)$
$={\log }_3\tan 1^o+{\log }_3\tan 2^o+{\log }_3\tan 3^o.........+{\log }_3\tan {89}^o$
$={\log }_3(\tan 1^o\times \tan 2^o\times ............\times \tan {89}^o),[\because \log a+\log b+....=\log (ab.......\left)\right]$
$=\log \left[\right(\tan 1^o\times \tan {89}^o\left)\right(\tan 2^o\times \tan {88}^o)...(\tan {44}^o\times \tan {46}^o\left)\right(\tan {45}^o\left)\right]$
$\Rightarrow S={\log }_{3}1=0$