A) -1
B) 1
C) 0
D) None of these
Correct Answer: A
Solution :
\[{{({{\tan }^{-1}}x)}^{2}}+{{({{\cot }^{-1}}x)}^{2}}=\frac{5{{\pi }^{2}}}{8}\] Þ \[{{({{\tan }^{-1}}x+{{\cot }^{-1}}x)}^{2}}-2{{\tan }^{-1}}x\left( \frac{\pi }{2}-{{\tan }^{-1}}x \right)=\frac{5{{\pi }^{2}}}{8}\] Þ \[\frac{{{\pi }^{2}}}{4}-2\times \frac{\pi }{2}{{\tan }^{-1}}x+2{{({{\tan }^{-1}}x)}^{2}}=\frac{5{{\pi }^{2}}}{8}\] Þ \[2{{({{\tan }^{-1}}x)}^{2}}-\pi {{\tan }^{-1}}x-\frac{3{{\pi }^{2}}}{8}=0\] Þ \[{{\tan }^{-1}}x=-\frac{\pi }{4},\frac{3\pi }{4}\]Þ \[{{\tan }^{-1}}x=-\frac{\pi }{4}\Rightarrow x=-1\].
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