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

The general solution for cos(2x)=1-sin(x) is x=2pin,x=pi+2pin,x= pi/6+2pin,x=(5pi)/6+2pin

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

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

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

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

Subtract

1 - sin (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

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

cos (2 x) + 3 sin (x) + 1 = 0

cos (z) = 2

cos (z) = 3

What is the general solution for cos(2x)=1-sin(x) ?
The general solution for cos(2x)=1-sin(x) is x=2pin,x=pi+2pin,x= pi/6+2pin,x=(5pi)/6+2pin