Question

Solve the equation $:\cos x+\cos 3x-2\cos 2x=0$

Answer 1

$cosx+(4co{s}^{3}x-3cosx)-2(2co{s}^{2}x-1)=0$

$cosx+4co{s}^{3}x-3cosx-4co{s}^{2}x+2=0$

$4co{s}^{3}x-4co{s}^{2}x-2cosx+2=0$

$4co{s}^{2}x(cosx-1)-2(cosx-1)=0$

$(4co{s}^{2}x-2)(cosx-1)=0$

$(2co{s}^{2}x-1)(cosx-1)=0$

$\therefore co{s}^{2}x=1/2$

or $co{s}^{2}x=co{s}^2\left(\pi /3\right)$

$\therefore x=x\pi \pm \pi /3$

and if $cosx-1=0$

$cosx=1$

$cosx=cos0$

$\therefore x=2x\pi$