Multiply through by cos(x):
1+2cos2(x)=4cos(x),
2cos2(x)-4cos(x)+1=0, (quadratic in cos(x))
cos2(x)-2cos(x)+½=0, (dividing through by 2)
cos2(x)-2cos(x)=-½,
cos2(x)-2cos(x)+1=1-½=½, (completing the square)
(cos(x)-1)2=½,
cos(x)-1=±√½, (taking the square root of both sides)
cos(x)=1±√½. cosine cannot exceed 1 or be less than -1, so cos(x)=1-√½.
x=±72.97°+360n° where n is an integer. x=±1.2735+2πn radians.