A) 5
B) 6
C) 7
D) None of these
Correct Answer: C
Solution :
\[\cot \left( \frac{\pi }{4}-2.{{\cot }^{-1}}3 \right)\] \[=\cot \left( \frac{\pi }{4}-2{{\tan }^{-1}}\left( \frac{1}{3} \right) \right)=\cot \left( \frac{\pi }{4}-{{\tan }^{-1}}\left( \frac{2/3}{1-1/9} \right) \right)\]\[=\cot \left( \frac{\pi }{4}-{{\tan }^{-1}}\left( \frac{2/3}{8/9} \right) \right)=\cot \left( \frac{\pi }{4}-{{\tan }^{-1}}\left( \frac{2/3}{8/9} \right) \right)\]\[=\cot \left( {{\tan }^{-1}}1-{{\tan }^{-1}}\left( \frac{3}{4} \right) \right)=\cot \left( {{\tan }^{-1}}\left( \frac{1-3/4}{1+3/4} \right) \right)\]\[=\cot \left( {{\tan }^{-1}}\left( \frac{1/4}{7/4} \right) \right)=\cot \left( {{\tan }^{-1}}\left( \frac{1}{7} \right) \right)\] \[=\cot \left( {{\cot }^{-1}}(7) \right)=7\] Hence, option is correct.
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