Question

Write the value of ${\tan }^{-1}\left(\frac{1}{x}\right)$ for x < 0 in terms of ${\cot }^{-1}x$

Answer 1

$$\begin{gathered} {\tan }^{-1}\left(\frac{1}{x}\right)={\tan }^{-1}\left(-\frac{1}{x}\right)\mathrm{for}x<0 \\ =-{\tan }^{-1}\left(\frac{1}{x}\right) \\ =-{\cot }^{-1}x \\ =-\left(\mathrm{\pi }-{\cot }^{-1}x\right) \\ =-\mathrm{\pi }+{\cot }^{-1}x \end{gathered}$$