<s>[INST] Why does the following equality hold? $\sec^{-1}(2/\sqrt{2}) = \sec^{-1}(\sqrt{2})$? Why is $\sec^{-1}(2/\sqrt{2}) = \sec^{-1}(\sqrt{2})$ true?
 [/INST] As Workaholic pointed out (here with more explanation):
\begin{align}
\frac{2}{\sqrt{2}} &= \frac{2}{\sqrt{2}}\cdot \frac{\sqrt{2}}{\sqrt{2}}\tag{$\frac{\sqrt{2}}{\sqrt{2}}=1$}\\[1em]
                   &= \frac{2\sqrt{2}}{2}\\[1em]
                   &= \sqrt{2}.
\end{align}
Since $\dfrac{2}{\sqrt{2}}=\sqrt{2}$, it must then be true that
$$
\sec^{-1}\left(\frac{2}{\sqrt{2}}\right)=\sec^{-1}(\sqrt{2}),
$$
as desired.
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