Given the equation:
[tex]\cos \mleft(2x\mright)-cos\mleft(x\mright)=0[/tex]First, we will express the cos (2x) in terms of cos (x) using the angle double rule:
[tex]\cos (2x)=2\cos ^2(x)-1[/tex]So, the given equation will be:
[tex]\begin{gathered} 2\cos ^2(x)-1-\cos (x)=0 \\ 2\cos ^2(x)-\cos (x)-1=0 \end{gathered}[/tex]The last equation has the form of the quadratic equation, so we will factor the equation:
[tex]\begin{gathered} (2\cos +1)(\cos x-1)=0 \\ 2\cos x+1=0\rightarrow\cos x=-\frac{1}{2}\rightarrow x=\cos ^{-1}(-\frac{1}{2})=120\degree \\ \cos x-1=0\rightarrow\cos x=1\rightarrow x=\cos ^{-1}(1)=0\degree \end{gathered}[/tex]So, the answer will be option 3) x = 0° or 120°
