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

The general solution for cos(2x)-sin(x)=1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin,x=2pin,x=pi+2pin

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

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

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = 2 πn, x = π + 2 πn

Solution steps

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

Subtract

1

from both sides

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

Rewrite using trig identities

- sin (x) - 2 sin^{2} (x) = 0

Solve by substitution

sin (x) = - \frac{1}{2}, sin (x) = 0

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn, x = 2 πn, x = π + 2 πn

sec (3 x) = - 1

2 cos^{2} (x) + 1 = 3 cos (x)

cos (x) + 2 = 3 cos (x)

tan (2 θ) = 0

9 tan (x) sin (x) = - 8 sin (x)

What is the general solution for cos(2x)-sin(x)=1 ?
The general solution for cos(2x)-sin(x)=1 is x=(7pi)/6+2pin,x=(11pi)/6+2pin,x=2pin,x=pi+2pin