The value of $ an ^{-1}\left(\frac{1}{3} ight)+ an ^{-1}\left(\frac{1}{5} ight)+ an ^{-1}\left(\frac{1}{7} ight)+ an ^{-1}\left(\frac{1}{8} ight)$ is

Question

The value of $	an ^{-1}\left(\frac{1}{3}ight)+	an ^{-1}\left(\frac{1}{5}ight)+	an ^{-1}\left(\frac{1}{7}ight)+	an ^{-1}\left(\frac{1}{8}ight)$ is

Aicharya · Accepted Answer

$\begin{aligned} & 	an ^{-1}\left(\frac{1}{3}ight)+	an ^{-1}\left(\frac{1}{5}ight)+	an ^{-1}\left(\frac{1}{7}ight)+	an ^{-1}\left(\frac{1}{8}ight) \ & =	an ^{-1}\left(\frac{\frac{1}{3}+\frac{1}{5}}{1-\frac{1}{3} 	imes \frac{1}{5}}ight)+	an ^{-1}\left(\frac{\frac{1}{7}+\frac{1}{8}}{1-\frac{1}{7} 	imes \frac{1}{8}}ight) \ & =	an ^{-1}\left(\frac{4}{7}ight)+	an ^{-1}\left(\frac{3}{11}ight) \ & =	an ^{-1} \frac{\frac{4}{7}+\frac{3}{11}}{\left(1-\frac{4}{7} 	imes \frac{3}{11}ight)}=	an ^{-1}\left(\frac{65}{65}ight)=	an ^{-1}(1)=\frac{\pi}{4}\end{aligned}$

The value of $\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan…

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