Question

how can we prove that cot inverse (-x) = pie - cot inverse (x)?

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ {\cot }^{-1}\left(-\mathrm{x}\right)=\mathrm{A} \\ -\mathrm{x}=\mathrm{cotA} \\ \mathrm{x}=-\mathrm{cotA}=\cot \left(\mathrm{\pi }-\mathrm{A}\right) \\ \mathrm{\pi }-\mathrm{A}={\cot }^{-1}\mathrm{x} \\ \mathrm{A}=\mathrm{\pi }-{\cot }^{-1}\mathrm{x} \\ {\cot }^{-1}\left(-\mathrm{x}\right)=\mathrm{\pi }-{\cot }^{-1}\mathrm{x} \\ \mathrm{Regards} \end{gathered}$$