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

The general solution for 2sin(x/2)=1-cos(x) is x=4pin,x=2pi+4pin,x=pi+4pin

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

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

x = 4 πn, x = 2 π + 4 πn, x = π + 4 πn

Solution steps

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

Subtract

1 - cos (x)

from both sides

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

Let:

u = \frac{x}{2}

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

Rewrite using trig identities

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

Solve by substitution

sin (u) = 0, sin (u) = 1

sin (u) = 0 : u = 2 πn, u = π + 2 πn

sin (u) = 1 : u = \frac{π}{2} + 2 πn

Combine all the solutions

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

Substitute back

u = \frac{x}{2}

\frac{x}{2} = 2 πn : x = 4 πn

\frac{x}{2} = π + 2 πn : x = 2 π + 4 πn

\frac{x}{2} = \frac{π}{2} + 2 πn : x = π + 4 πn

x = 4 πn, x = 2 π + 4 πn, x = π + 4 πn

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

12 sin (A) + 4 = 4 sin (A) + 4

2 sin^{2} (x) = 3 - 3 cos (x)

cos (x) + sin (x) tan (x) = 1

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

What is the general solution for 2sin(x/2)=1-cos(x) ?
The general solution for 2sin(x/2)=1-cos(x) is x=4pin,x=2pi+4pin,x=pi+4pin