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

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

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

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

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

Solution steps

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

Subtract

cos (2 x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

sin (x) = 1, sin (x) = 0

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 0 : x = 2 πn, x = π + 2 πn

Combine all the solutions

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

tan (\frac{x}{2}) = \frac{1}{3}

6 cos (x) = 5

6 cos (2 x) = 6 cos^{2} (x) - 5

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

tan (x) = 1, - 2 π \leq x \leq 2 π

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