The set of all values of k for which ${\left({\tan }^{-1}\mathrm{x}\right)}^3+{\left({\cot }^{-1}\mathrm{x}\right)}^3=\mathrm{k}{\pi }^3,\mathrm{x}\in \mathrm{R}$  is the interval :

Let S = ${\left({\tan }^{-1}\mathrm{x}\right)}^3+{\left({\cot }^{-1}\mathrm{x}\right)}^3$
$$\begin{gathered} \begin{array}{l}={\left({\tan }^{-1}\mathrm{x}+{\cot }^{-1}\mathrm{x}\right)}^3-3{\tan }^{-1}\mathrm{x}\cdot {\cot }^{-1}\mathrm{x}\left({\tan }^{-1}\mathrm{x}+{\cot }^{-1}\mathrm{x}\right)\\ =\frac{{\mathrm{\pi }}^3}{8}-\frac{3\mathrm{\pi }}{2}{\tan }^{-1}\mathrm{x}\left(\frac{\mathrm{\pi }}{2}-{\tan }^{-1}\mathrm{x}\right)\\ =\frac{3\mathrm{\pi }}{2}{\left({\tan }^{-1}\mathrm{x}-\frac{\mathrm{\pi }}{4}\right)}^2+\frac{{\mathrm{\pi }}^3}{32}\\ \Rightarrow \frac{{\mathrm{\pi }}^3}{32}\le \mathrm{S}<\frac{7}{8}{\mathrm{\pi }}^3 \\ \begin{array}{l}=\frac{{\mathrm{\pi }}^3}{32}\le {\mathrm{K\pi }}^3<\frac{7}{8}{\mathrm{\pi }}^3\\ \frac{1}{32}\le \mathrm{K}<\frac{7}{8}\end{array}\end{array} \end{gathered}$$