establish identity 1-cos(2x)=tan(x)sin(2x)

Question

establish identity

 

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

 

(1+cosθ)tan θ/2 =sinθ

Answer 1

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

working with the left side first

1-cos(2x)

1-(cos²(x)-sin²(x))

[1-cos²(x)]+sin²x

sin²(x)+sin²(x)

2sin²(x) is the left side

working with the right side

tan(x)(sin(2x))

(sin[x]/cos[x])(2sin(x)cos(x))

sin(x)(2sin(x))   (the 2 cos(x)'s cancel)

2sin²(x) is the right side also so it has been established

 

(1+cos(x))(tan(x/2)=sin(x)    (I'm using x instead of θ)

(1+cos(x))(√(1-cos[x])/(1+cos[x])=sin(x)

(√(1-cos[x])/(1+cos[x])=sin(x)/(1+cos[x])

square both sides

(1-cos[x])/(1+cos[x])=sin²(x)/(1+cos[x])²

cross multiply

(1-cos[x])(1+cos[x])(1+cos[x])=(1+cos[x])(sin²(x))

divide both sides by (1+cos[x])

(1-cos[x])(1+cos[x])=sin²(x)

1-cos²(x)=sin²(x)

1=sin²(x)+cos²(x) which is a trig identity so identity established