Considering only the principal values of the inverse trigonometric functions, the value of  $\frac{3}{2}{\cos }^{-1}\sqrt{\frac{2}{2+{\pi }^2}}+\frac{1}{4}{\sin }^{-1}\frac{2\sqrt{2}\pi }{2+{\pi }^2}+{\tan }^{-1}\frac{\sqrt{2}}{\pi }$  is _____.

${\cos }^{-1}\sqrt{\frac{2}{2+{\pi }^2}}={\tan }^{-1}\frac{\pi }{\sqrt{2}}$ 
 
 ${\sin }^{-1}\left(\frac{2\sqrt{2}\pi }{2+{\pi }^2}\right)={\sin }^{-1}\left(\frac{2\times \frac{\pi }{\sqrt{2}}}{1+{\left(\frac{\pi }{\sqrt{2}}\right)}^2}\right)$
 $=\pi -2{\tan }^{-1}\left(\frac{\pi }{\sqrt{2}}\right)$
$(\text{As},{\sin }^{-1}\left(\frac{2x}{1+x^2}\right)=\pi -2{\tan }^{-1}x,x\ge 1)$ 
and  ${\tan }^{-1}\frac{\sqrt{2}}{\pi }={\cot }^{-1}\left(\frac{\pi }{\sqrt{2}}\right)$
$\therefore$  Expression  $=\frac{3}{2}\left({\tan }^{-1}\frac{\pi }{\sqrt{2}}\right)+\frac{1}{4}(\pi -2{\tan }^{-1}\frac{\pi }{\sqrt{2}})+{\cot }^{-1}\left(\frac{\pi }{\sqrt{2}}\right)$
 $$\begin{gathered} =(\frac{3}{2}-\frac{2}{4}){\tan }^{-1}\frac{\pi }{\sqrt{2}}+\frac{\pi }{4}+{\cot }^{-1}\frac{\pi }{\sqrt{2}} \\ =({\tan }^{-1}\frac{\pi }{\sqrt{2}}+{\cot }^{-1}\frac{\pi }{\sqrt{2}})+\frac{\pi }{4} \\ =\frac{\pi }{2}+\frac{\pi }{4}=\frac{3\pi }{4} \end{gathered}$$
= 2.35 or 2.36