If the roots of the equation ${\mathrm{x}}^3-10\mathrm{x}+11=0$ are u, v, w then the value of $3\mathrm{cosec}{}^2\left({\tan }^{-1}\mathrm{u}+{\tan }^{-1}\mathrm{v}+{\tan }^{-1}\mathrm{w}\right)$ is

u, v, w are the roots of ${\mathrm{x}}^3-10\mathrm{x}+11=0$ then $\mathrm{u}+\mathrm{v}+\mathrm{w}=0, \mathrm{uv}+\mathrm{vw}+\mathrm{wu}=-10, \mathrm{uvw}=-11$

let $\mathrm{u}=\tan \mathrm{\alpha },\mathrm{v}=\tan \mathrm{\beta },\mathrm{w}=\tan \mathrm{\gamma }$

$\begin{array}{rcl}\tan (\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })& =& \frac{\sum \tan \mathrm{\alpha }-\mathrm{\pi tan}\mathrm{\alpha }}{1-\sum \tan \mathrm{\alpha tan}\mathrm{\beta }}\\ & =& \frac{0+11}{1+10}=\frac{11}{11}=1=\tan \frac{\mathrm{\pi }}{4}\end{array}$

$$\begin{gathered} \mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma }=\frac{\mathrm{\pi }}{4} \\ 3\mathrm{cosec}{}^2\left({\tan }^{-1}\mathrm{u}+{\tan }^{-1}\mathrm{v}+{\tan }^{-1}\mathrm{w}\right) \\ =3\mathrm{cosec}{}^2(\mathrm{\alpha }+\mathrm{\beta }+\mathrm{\gamma })=3{\mathrm{cosec}}^2\frac{\mathrm{\pi }}{4}=3(\sqrt{2}{)}^2=3\times 2=6 \end{gathered}$$