What is the general solution for 2cos^2(x)-cos(x)=2-sec(x) ?

The general solution for 2cos^2(x)-cos(x)=2-sec(x) is x= pi/3+2pin,x=(5pi)/3+2pin,x=pi+2pin,x=2pin

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2cos^2(x)-cos(x)=2-sec(x)

2 cos^{2} (x) - cos (x) = 2 - sec (x)

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn, x = 2 πn

Solution steps

2 cos^{2} (x) - cos (x) = 2 - sec (x)

Subtract

2 - sec (x)

from both sides

2 cos^{2} (x) - cos (x) - 2 + sec (x) = 0

Rewrite using trig identities

- 2 - \frac{1}{sec ( x )} + \frac{2}{sec ^{2} ( x )} + sec (x) = 0

Solve by substitution

sec (x) = 2, sec (x) = - 1, sec (x) = 1

sec (x) = 2 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

sec (x) = - 1 : x = π + 2 πn

sec (x) = 1 : x = 2 πn

Combine all the solutions

x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn, x = π + 2 πn, x = 2 πn

sin^{4} (x) - cos^{4} (x) = \frac{1}{2}

x, y = tan (x)

sin (θ) = - 2

\frac{cos ( \frac{x}{2} )}{2} = 0

sin (2 x) = cos (3 x)

What is the general solution for 2cos^2(x)-cos(x)=2-sec(x) ?
The general solution for 2cos^2(x)-cos(x)=2-sec(x) is x= pi/3+2pin,x=(5pi)/3+2pin,x=pi+2pin,x=2pin