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

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

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

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

x = \frac{π}{2} + 4 πn, x = 4 π - \frac{π}{2} + 4 πn

Solution steps

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

Subtract

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

from both sides

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

Let:

u = \frac{x}{2}

cos (u) - \frac{1 - cos ( 2 u )}{2} = 0

Rewrite using trig identities

cos (u) - 1 - cos^{2} (u) = 0

Solve by substitution

cos (u) = \frac{1}{2}

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

Combine all the solutions

u = arccos (\frac{1}{2}) + 2 πn, u = 2 π - arccos (\frac{1}{2}) + 2 πn

Substitute back

u = \frac{x}{2}

\frac{x}{2} = arccos (\frac{1}{2}) + 2 πn : x = \frac{π}{2} + 4 πn

\frac{x}{2} = 2 π - arccos (\frac{1}{2}) + 2 πn : x = 4 π - \frac{π}{2} + 4 πn

x = \frac{π}{2} + 4 πn, x = 4 π - \frac{π}{2} + 4 πn

0.322 = sin (3.92232 \cdot x)

x, arcsin (y) = arcsin (x)

x, cos (2 x) = \frac{1}{2}

(10710 \cdot cos (2 \cdot a)) + 336.22 = 0

tan (θ) = \frac{2}{- 2 3}

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