A) \[\frac{\pi }{2}\]
B) \[\frac{\pi }{3}\]
C) \[\frac{\pi }{4}\]
D) None of these
Correct Answer: C
Solution :
\[4{{\tan }^{-1}}\frac{1}{5}-{{\tan }^{-1}}\frac{1}{70}+{{\tan }^{-1}}\frac{1}{99}\] \[=2{{\tan }^{-1}}\left[ \frac{\frac{2}{5}}{1-\frac{1}{25}} \right]-{{\tan }^{-1}}\frac{1}{70}+{{\tan }^{-1}}\frac{1}{99}\] \[=2{{\tan }^{-1}}\left( \frac{5}{12} \right)-{{\tan }^{-1}}\frac{1}{70}+{{\tan }^{-1}}\frac{1}{99}\] \[={{\tan }^{-1}}\left[ \frac{\frac{5}{6}}{1-\frac{25}{144}} \right]-{{\tan }^{-1}}\frac{1}{70}+{{\tan }^{-1}}\frac{1}{99}\] \[={{\tan }^{-1}}\left( \frac{120}{119} \right)-{{\tan }^{-1}}\frac{1}{70}+{{\tan }^{-1}}\frac{1}{99}\] \[={{\tan }^{-1}}\left( \frac{120}{119} \right)+{{\tan }^{-1}}\left[ \frac{\frac{1}{99}-\frac{1}{70}}{1+\frac{1}{99}.\frac{1}{70}} \right]\] \[={{\tan }^{-1}}\left( \frac{120}{119} \right)+{{\tan }^{-1}}\left( \frac{-29}{6931} \right)\] \[={{\tan }^{-1}}\frac{120}{119}-{{\tan }^{-1}}\frac{29}{6931}={{\tan }^{-1}}\frac{120}{119}-{{\tan }^{-1}}\frac{1}{239}\] \[={{\tan }^{-1}}\left[ \frac{\frac{120}{119}-\frac{1}{239}}{1+\frac{120}{119}\times \frac{1}{239}} \right]={{\tan }^{-1}}(1)=\frac{\pi }{4}\].
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