A) [a] \[ta{{n}^{-1}}({{x}^{2}}+x+1)\]
B) (b)\[co{{t}^{-1}}({{x}^{2}}-x+1)\]
C) \[co{{t}^{-1}}({{x}^{2}}+x+1)\]
D) \[ta{{n}^{-1}}({{x}^{2}}-x+1)\]
Correct Answer: A
Solution :
[a] \[y={{\tan }^{-1}}x+{{\cot }^{-1}}(x+1)\] \[={{\tan }^{-1}}x+{{\tan }^{-1}}\left( \frac{1}{x+1} \right)\] \[={{\tan }^{-1}}\left( \frac{x+\frac{1}{x+1}}{1-\frac{1}{x+1}.x} \right)={{\tan }^{-1}}\left( \frac{{{x}^{2}}+x+1}{1+x-x} \right)\] \[=ta{{n}^{-1}}({{x}^{2}}+x+1)\] Hence, option [a] is correct.
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