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 __________ .

The correct answer is 2.36 (which is equivalent to $\frac{3\pi }{4}$).

Let us evaluate the expression step-by-step considering only the principal values of inverse trigonometric functions.
The given expression is:

$E=\frac{3}{2}{\cos }^{-1}\left(\sqrt{\frac{2}{2+{\pi }^2}}\right)+\frac{1}{4}{\sin }^{-1}\left(\frac{2\sqrt{2}\pi }{2+{\pi }^2}\right)+{\tan }^{-1}\left(\frac{\sqrt{2}}{\pi }\right)$

Let $\theta ={\tan }^{-1}\left(\frac{\sqrt{2}}{\pi }\right)$.
Since $\frac{\sqrt{2}}{\pi }>0$, $\theta \in \left(0,\frac{\pi }{2}\right)$.
From $\tan \theta =\frac{\sqrt{2}}{\pi }$, we can draw a right-angled triangle with:
• Opposite side = $\sqrt{2}$
• Adjacent side = $\pi$
• Hypotenuse = $\sqrt{{\left(\sqrt{2}\right)}^2+{\pi }^2}=\sqrt{2+{\pi }^2}$

Using this right triangle, we can write:

$\cos \theta =\frac{\pi }{\sqrt{2+{\pi }^2}}$ and $\sin \theta =\frac{\sqrt{2}}{\sqrt{2+{\pi }^2}}$

Now, let us rewrite each term in the expression in terms of $\theta$:

First Term:
${\cos }^{-1}\left(\sqrt{\frac{2}{2+{\pi }^2}}\right)={\cos }^{-1}\left(\sin \theta \right)={\cos }^{-1}\left(\cos \left(\frac{\pi }{2}-\theta \right)\right)=\frac{\pi }{2}-\theta$

Second Term:
Observe that:

$\sin \left(2\theta \right)=2\sin \theta \cos \theta =2·\left(\frac{\sqrt{2}}{\sqrt{2+{\pi }^2}}\right)·\left(\frac{\pi }{\sqrt{2+{\pi }^2}}\right)=\frac{2\sqrt{2}\pi }{2+{\pi }^2}$

Since $0<\theta <\frac{\pi }{2}$ and $\tan \theta =\frac{\sqrt{2}}{\pi }<1$, we have $0<\theta <\frac{\pi }{4}$, which implies $0<2\theta <\frac{\pi }{2}$.
Therefore:

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

Now, substituting these simplified terms back into the expression $E$:

$E=\frac{3}{2}\left(\frac{\pi }{2}-\theta \right)+\frac{1}{4}\left(2\theta \right)+\theta$

Expanding and grouping the terms:

$E=\frac{3\pi }{4}-\frac{3}{2}\theta +\frac{1}{2}\theta +\theta$

$E=\frac{3\pi }{4}+\left(-\frac{3}{2}+\frac{1}{2}+1\right)\theta$

$E=\frac{3\pi }{4}+0·\theta =\frac{3\pi }{4}$

Substituting $\pi \approx 3.14159$:

$E=\frac{3\times 3.14159}{4}\approx 2.35619\approx 2.36$

Thus, the value of the given expression rounded to two decimal places is 2.36.