Question

sec(x)+2cos(x)=4

Answer 1

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.