A) \[\frac{1}{3}\]
B) \[\frac{1}{2}\]
C) \[\frac{1}{4}\]
D) \[\frac{1}{6}\]
Correct Answer: B
Solution :
Given equation \[{{\tan }^{-1}}x=\frac{\pi }{4}-{{\tan }^{-1}}\left( \frac{1}{3} \right)\] \[\Rightarrow \] \[{{\tan }^{-1}}x={{\tan }^{-1}}(1)-{{\tan }^{-1}}\left( \frac{1}{3} \right)\] \[\left( \because \,\,\frac{\pi }{4}={{\tan }^{-1}}(1) \right)\] \[\Rightarrow \] \[{{\tan }^{-1}}x={{\tan }^{-1}}\left( \frac{1-\frac{1}{3}}{1+\frac{1}{3}} \right)\] \[\left[ \because \,\,{{\tan }^{-1}}x-{{\tan }^{-1}}y={{\tan }^{-1}}\left( \frac{x-y}{1+xy} \right) \right]\] \[\Rightarrow \] \[{{\tan }^{-1}}x={{\tan }^{-1}}\left( \frac{2/3}{4/3} \right)\] \[\Rightarrow \] \[{{\tan }^{-1}}x={{\tan }^{-1}}(1/2)\] \[\Rightarrow \] \[x=1/2\]
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