The value of $sin \left[cot^{-1}\begin{Bmatrix}cos(tan^{-1}1)\end{Bmatrix}\right]$, is

$\sqrt{\frac{2}{3}}$

$sin \left[cot^{-1}\begin{Bmatrix}cos(tan^{-1}1)\end{Bmatrix}\right]$

$=sin\begin{Bmatrix}cot^{-1}\left(cos\frac{\pi}{4}\right)\end{Bmatrix}$   $\left[∵tan^{-1}1=\frac{\pi}{4}\right]$

$=sin\left(cot^{-1}\frac{1}{\sqrt{2}}\right)$ 

  $=sin\left(sin^{-1}\frac{\sqrt{2}}{\sqrt{3}}\right)=\frac{\sqrt{2}}{\sqrt{3}}=\sqrt{\frac{2}{3}}$         $\left[∵cot^{-1}\frac{1}{\sqrt{2}}=sin^{-1}\frac{\sqrt{2}}{\sqrt{3}}\right]$