What is the general solution for sec(x)-2cos(x)=1 ?

The general solution for sec(x)-2cos(x)=1 is x= pi/3+2pin,x=(5pi)/3+2pin,x=pi+2pin

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

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

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

Solution steps

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

Subtract

1

from both sides

sec (x) - 2 cos (x) - 1 = 0

Rewrite using trig identities

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

Solve by substitution

sec (x) = 2, 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

Combine all the solutions

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

2 sin (x) + 3 = 0

4 cos (2 θ) - 10 cos (θ) + 14 = 7

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

0 = 8 cos (x) - 8 sin (x)

sec (8 x) = 2

What is the general solution for sec(x)-2cos(x)=1 ?
The general solution for sec(x)-2cos(x)=1 is x= pi/3+2pin,x=(5pi)/3+2pin,x=pi+2pin