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

The general solution for (sin(x))/(1+cos(x))=sin(x/2) is x=4pin,x=2pi+4pin,x=pi+4pin,x=3pi+4pin

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

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

x = 4 πn, x = 2 π + 4 πn, x = π + 4 πn, x = 3 π + 4 πn

Solution steps

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

Subtract

sin (\frac{x}{2})

from both sides

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

Simplify

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Let:

u = \frac{x}{2}

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

Rewrite using trig identities

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

Factor

- (1 + cos (2 u)) sin (u) + 2 cos (u) sin (u) : sin (u) (- 1 - cos (2 u) + 2 cos (u))

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

Solving each part separately

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

sin (u) = 0 : u = 2 πn, u = π + 2 πn

- 1 - cos (2 u) + 2 cos (u) = 0 : u = \frac{π}{2} + 2 πn, u = \frac{3 π}{2} + 2 πn, u = 2 πn

Combine all the solutions

u = 2 πn, u = π + 2 πn, u = \frac{π}{2} + 2 πn, u = \frac{3 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

\frac{x}{2} = \frac{3 π}{2} + 2 πn : x = 3 π + 4 πn

x = 4 πn, x = 2 π + 4 πn, x = π + 4 πn, x = 3 π + 4 πn

Since the equation is undefined for:

π + 4 πn, 3 π + 4 πn

x = 4 πn, x = 2 π + 4 πn, x = π + 4 πn, x = 3 π + 4 πn

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

2 sin (θ) cos (θ) + 2 sin (θ) - 2 cos (θ) sin (θ) = 0

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

cosh (x) = \frac{3}{2}

\frac{1 + tan ( t )}{sin ( t )} = 0

What is the general solution for (sin(x))/(1+cos(x))=sin(x/2) ?
The general solution for (sin(x))/(1+cos(x))=sin(x/2) is x=4pin,x=2pi+4pin,x=pi+4pin,x=3pi+4pin