write the values of x for which tan^-1 (1/x)=cot^-1 x holds

Dear Student,

$$\begin{gathered} let {\tan }^{-1}\left(\frac{1}{x}\right)=A.........................................\left(1\right) \\ and co{t}^{-1}x=B...........................................\left(2\right) \\ From \left(1\right), we have \\ {\tan }^{-1}\left(\frac{1}{x}\right)=A \\ \Rightarrow \frac{1}{x}=\tan A \\ \Rightarrow \tan A=\frac{1}{x}=\frac{Perpendicular}{Base} \\ \Rightarrow cot A=\frac{Base}{Perpendicular}=\frac{x}{1}=x \\ \Rightarrow cot A=x \\ \Rightarrow A=co{t}^{-1}x \\ but co{t}^{-1}x=B (from (2\left)\right) \\ \Rightarrow A=B \\ \Rightarrow so, It is independent of the value of x \\ \Rightarrow \boxed{ x can take any real value or x\in R } \end{gathered}$$

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