Question

cos^2x = cos x solve each equation for principal values of x. express solutions in degrees

Answer 1

Simplify cos(2x) first.
\[\cos(2x)=\cos^{2}(x)-\sin^{2}(x)\]
\[=\cos^{2}(x)-(1-cos^{2}(x))\]
\[=\cos^{2}(x)-1+\cos^{2}(x)\]
\[=2\cos^{2}(x)-1\]
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\[2\cos^{2}(x)-1=\cos(x)\]
\[2\cos^{2}(x)-\cos(x)-1=0\]
\[(2\cos(x)+1)(\cos(x)-1)=0\]
\[\cos(x)=-\frac{ 1 }{ 2 }\]
or
\[\cos(x)=1\]
\[x=0, 120, 240, 360 \]