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

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

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

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

x = 4 πn, x = 2 π + 4 πn

Solution steps

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

Subtract

sin (x)

from both sides

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

Let:

u = \frac{x}{2}

2 sin (u) - sin (2 u) = 0

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (u) = 0 : u = 2 πn, u = π + 2 πn

cos (u) - 1 = 0 : u = 2 πn

Combine all the solutions

u = 2 πn, u = π + 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

x = 4 πn, x = 2 π + 4 πn

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

15 cos (x) - 5 = 4 cos (x)

sin (θ) + cos (θ) = \frac{2 + 3}{2}

x, arctan (\frac{1}{x}) = 8 0^{\circ}

2 sin (θ) cos (θ) + 2 sin (θ) + cos (θ) + 1 = 0

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