Question

Solve the following equation:
$\cos 4x+2{\cos }^{2}x=1.$

Answer 1

${2\cos }^{2}2x-1=1-{2\cos }^{2}x=-\cos 2x$

Let $\cos 2x=z.$

${2z}^2+z-1=0$

$(2z-1)(z+1)=0$

So, $\cos 2x=-1$ and $\cos 2x=\frac{1}{2}.$

$\therefore$ $x=(2n+1)\frac{\pi }{2}$ and $x=n\pi \pm \frac{\pi }{6}.$