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

Question

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

Answer 1

1-(cos²x/2-sin²x/2)= 2·sinx/2·cosx/2·sinx/2

 cos²x/2 + sin²x/2 - cos²x/2 + sin²x/2 = 2sin²x/2·cosx/2

2sin²x/2 - 2sin²x/2·cosx/2 = 0

2sin²x/2·( 1 - cosx/2) =0

sinx/2 = 0       или     1 - cosx/2 = 0

x/2 = πn, n∈Z,            cosx/2 = 1

х =  2πn, n∈Z,            x/2 = 2πk, k∈Z

                                    х=4πk, k∈Z.

Объединяя корни, получим, х =  2πn, n∈Z.

Ответ:  2πn, n∈Z

Answer 2

2sin^2x/2=2sin^2x/2cosx/2

sin^2x/2(1-cosx/2)=0

cosx/2=1

x/2=2пk

x=4Пk

sinx/2=0

x/2=Пk

x=2Пk

 

x=2Пk