[tex]\bf sec^2(x)-2=0\implies \cfrac{1^2}{cos^2(x)}-2=0\implies \cfrac{1}{cos^2(x)}=2
\\\\\\
\cfrac{1}{2}=cos^2(x)\implies \sqrt{\cfrac{1}{2}}=cos(x)
\\\\\\
cos^{-1}\left(\frac{1}{\sqrt{2}} \right)=cos^{-1}[cos(x)]\implies cos^{-1}\left(\frac{\sqrt{2}}{2} \right)=\measuredangle x
\\\\\\
\measuredangle x =
\begin{cases}
\frac{\pi }{4}\\\\
\frac{7\pi }{4}
\end{cases}[/tex]
now, those are the angles from [0, 2π], now, to include "all", using the "n" notation, for all coterminal angles
well, all you do, as before, add 2π or +2π to each
well.... 7π/4 is really -π/4 if you use negative angles
thus [tex]\bf x=\pm \frac{\pi }{4}+2\pi n \qquad \textit{ where "n" is an integer}[/tex]
