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

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

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

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

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

Solution steps

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

Subtract

2 cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

sin (x) = 1 : x = \frac{π}{2} + 2 πn

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

Combine all the solutions

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

cos (x) = 1 - cos (x)

sin (\frac{θ}{2}) = - \frac{1}{2}

sec^{2} (x) = 3 tan (x) + 1

- 3 cos (2 θ) - sin (θ) - 6 = - 8

sin^{2} (x) + 1 = \frac{7}{2} cos^{2} (x)

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