A) \[{{\cot }^{-1}}\left( \frac{1+\sqrt{2}}{1-\sqrt{2}} \right)\]
B) \[{{\cot }^{-1}}\left( \frac{\sqrt{2}+1}{\sqrt{2}-1} \right)\]
C) \[-\pi +{{\cot }^{-1}}\left( \frac{1+\sqrt{2}}{1-\sqrt{2}} \right)\]
D) \[\pi -{{\cot }^{-1}}\left( \frac{1-\sqrt{2}}{1+\sqrt{2}} \right)\]
Correct Answer: C
Solution :
[c] \[{{\tan }^{-1}}\left( \frac{1}{\sqrt{2}} \right)+{{\sin }^{-1}}\left( \frac{1}{\sqrt{5}} \right)-{{\cos }^{-1}}\left( \frac{1}{\sqrt{10}} \right)\] \[={{\tan }^{-1}}\left( \frac{1}{\sqrt{2}} \right)+{{\tan }^{-1}}\left( \frac{1}{2} \right)-{{\tan }^{-1}}3\] \[={{\tan }^{-1}}\left( \frac{1}{\sqrt{2}} \right)-\left[ {{\tan }^{-1}}3-{{\tan }^{-1}}\left( \frac{1}{2} \right) \right]\] \[={{\tan }^{-1}}\left( \frac{1}{\sqrt{2}} \right)-\left[ {{\tan }^{-1}}\left( \frac{3-\frac{1}{2}}{1+\frac{3}{2}} \right) \right]\] \[={{\tan }^{-1}}\left( \frac{1}{\sqrt{2}} \right)-{{\tan }^{-1}}1\] \[=-\left[ {{\tan }^{-1}}1-{{\tan }^{-1}}\left( \frac{1}{\sqrt{2}} \right) \right]=-\left[ {{\tan }^{-1}}\left( \frac{1-\frac{1}{\sqrt{2}}}{1+\frac{1}{\sqrt{2}}} \right) \right]\] \[=-{{\tan }^{-1}}\left( \frac{\sqrt{2}-1}{\sqrt{2}+1} \right)=-{{\cot }^{-1}}\left( \frac{\sqrt{2}+1}{\sqrt{2}-1} \right)\] \[=-\left[ {{\cot }^{-1}}\left\{ -\left( \frac{1+\sqrt{2}}{1-\sqrt{2}} \right) \right\} \right]=-\pi +{{\cot }^{-1}}\left( \frac{1+\sqrt{2}}{1-\sqrt{2}} \right)\]
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