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Answer:
$\frac{7\pi }{12}$
Hint:
$Separate\: and\: reorganize\: the\: values$
Given:
$Find\: principal\: value\: o\! f \sin^{-1}\left ( \frac{\sqrt{3}+1}{2\sqrt{2}} \right )$
Solution:
$\sin^{-1}\left ( \frac{\sqrt{3}+1}{2\sqrt{2}} \right )$$= \sin^{-1}\left ( \frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}} \right )$
$= \sin^{-1}\left ( \frac{\sqrt{3}}{2} \times \frac{1}{\sqrt{2}}+\frac{1}{2}\times \frac{1}{\sqrt{2}}\right )$
$= \sin^{-1}\left ( \frac{\sqrt{3}}{2} \times \sqrt{1-\left ( \frac{1}{\sqrt{2}} \right )^{2}}+\frac{1}{\sqrt{2}}\times \sqrt{1-\left ( \frac{\sqrt{3}}{2} \right )^{2}}\right )$
$= \sin^{-1}\left ( \frac{\sqrt{3}}{2} \right )+\sin^{-1}\left ( \frac{1}{\sqrt{2}} \right )$
$= \frac{\pi }{3}+\frac{\pi }{4}$
$= \frac{7\pi }{2}$
$Therefore, principal\: value\: o\! f \: \sin^{-1}\left ( \frac{\sqrt{3}+1}{2\sqrt{2}} \right ) is = \frac{7\pi }{2}.$

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