Let z= tan a/2 Show sin a= 2z/1+z2

Question

Let z= tan a/2  Show sin a= 2z/1+z²

Answer 1

z=tan(a/2) = √[(1 - cos(a))/(1 +cos(a))]

z²=(1 - cos(a))/(1 +cos(a))

z²(1+cos(a))=1 - cos(a)

z²-1=-cos(a)(1+z²)

cos(a)=(1-z²)/(1+z²)

 

sin²(a)=1-cos²(a)=1-[(1-z²)/(1+z²)]²=

=((1+z²)²-(1+z²)²)/(1+z²)²

=4z²/(1+z²)²

sin2(a)=2z/(1+z²)

 

=1