[tex]\bf sec(\theta)=\cfrac{1}{cos(\theta)}\\\\
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4cos(x)-2sec(x)=0\implies 4cos(x)-2\cfrac{1}{cos(x)}=0
\\\\\\
4cos(x)-\cfrac{2}{cos(x)}=0\implies \cfrac{4cos^2(x)-2}{cos(x)}=0
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4cos^2(x)-2=0\implies 4cos^2(x)=2\implies cos^2(x)=\cfrac{2}{4}[/tex]
[tex]\bf cos^2(x)=\cfrac{1}{2}\implies cos(x)=\pm\sqrt{\cfrac{1}{2}}\implies cos(x)=\pm\cfrac{\sqrt{1}}{\sqrt{2}}
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cos(x)=\pm\cfrac{1}{\sqrt{2}}\impliedby \textit{and rationalizing the denominator}
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cos(x)=\pm\cfrac{\sqrt{2}}{2}\implies \measuredangle x=
\begin{cases}
\frac{\pi }{4}\\\\
\frac{3\pi }{4}\\\\
\frac{5\pi }{4}\\\\
\frac{7\pi }{4}
\end{cases}[/tex]
