If ${\cos }^{-1}\mathrm{x}+{\cos }^{-1}\mathrm{y}+{\cos }^{-1}\mathrm{z}=\mathrm{\pi }\left[{\sec }^2\left(\mathrm{u}\right)+{\sec }^4\left(\mathrm{v}\right){\sec }^6\left(\mathrm{w}\right)\right]$ where $\mathrm{u},\mathrm{v},\mathrm{w}$ are least non-negative angles such that $\mathrm{u}<\mathrm{v}<\mathrm{w},$ then the value of ${\mathrm{x}}^{2000}+{\mathrm{y}}^{2000}+{\mathrm{z}}^{2000}+\frac{36\mathrm{\pi }}{\mathrm{u}+\mathrm{v}+\mathrm{w}}=$

$$\begin{gathered} {\sec }^2\mathrm{u},{\sec }^4\mathrm{v},{\sec }^6\in \left[1,\infty \right] \\ \therefore {\sec }^2\mathrm{u}+{\sec }^4\mathrm{v}+{\sec }^6\mathrm{we}\left[3,\infty \right] \\ \mathrm{\pi }\left({\sec }^2\mathrm{u}+{\sec }^4\mathrm{v}+{\sec }^6\mathrm{w}\right)\in \left[3\mathrm{\pi },\infty \right] \dots \dots \left(1\right) \end{gathered}$$

From (1) & (2) is L.H.S=R.H.S$=3\mathrm{\pi }$

$\therefore {\cos }^{-1}\mathrm{x}={\cos }^{-1}\mathrm{y}={\cos }^{-1}\mathrm{z}=\mathrm{\pi }$

$\therefore \mathrm{x}=\mathrm{y}=\mathrm{z}=-1$ and ${\sec }^2\mathrm{u}={\sec }^4\mathrm{v}={\sec }^6\mathrm{w}=1$

$$\begin{gathered} \therefore \mathrm{u}=\mathrm{\pi }, \mathrm{v}=2\mathrm{\pi }, \mathrm{w}=3\mathrm{\pi } \\ {\mathrm{x}}^{2000}+{\mathrm{y}}^{2000}+{\mathrm{z}}^{2000}+\frac{36\mathrm{\pi }}{\mathrm{u}+\mathrm{v}+\mathrm{w}}=1+1+1+\frac{36\mathrm{\pi }}{6\mathrm{\pi }}=9 \end{gathered}$$