Please find easy solution for this question
 

Dear Student,
Please find below the solution to the asked query:

$$\begin{gathered} {\sin }^{-1}\mathrm{x}+{\sin }^{-1}\mathrm{y}+{\sin }^{-1}\mathrm{z}=\frac{3\mathrm{\pi }}{2} \\ \mathrm{This} \mathrm{is} \mathrm{satisfied} \mathrm{only} \mathrm{when} \mathrm{x}=\mathrm{y}=\mathrm{z}=1, \\ {\sin }^{-1}\mathrm{x} \mathrm{has} \mathrm{range} \left[-\frac{\mathrm{\pi }}{2},\frac{\mathrm{\pi }}{2}\right] \\ \mathrm{hence}, \\ \frac{{\mathbf{x}}^{\mathbf{2006}}\mathbf{+}{\mathbf{y}}^{\mathbf{2007}}\mathbf{+}{\mathbf{z}}^{\mathbf{2008}}\mathbf{-}\mathbf{9}}{{\mathbf{x}}^{\mathbf{2006}}\mathbf{+}{\mathbf{y}}^{\mathbf{2007}}\mathbf{+}{\mathbf{z}}^{\mathbf{2008}}}\mathbf{=}\frac{\mathbf{3}\mathbf{-}\mathbf{9}}{\mathbf{3}}\mathbf{=}\frac{\mathbf{-}\mathbf{6}}{\mathbf{3}}\mathbf{=}\mathbf{-}\mathbf{2} \end{gathered}$$

Regards