2cos(x)+2=sec(x),
2cos2(x)+2cos(x)=1,
cos2(x)+cos(x)=½,
cos2(x)+cos(x)+¼=¾,
(cos(x)+½)2=¾,
cos(x)+½=±√3/2,
cos(x)=-½±√3/2; x=1.1961radians approx. (cosine must lie between -1 and 1, so other "solution" is rejected.)
x=-1.1961radians, so the more general answer is 2πn±1.1961 radians, where n is an integer.
This can also be written 360n°±68.5293°.