Question

Prove that (1-cosx+sinx)/(1+cosx + sinx) = tan(x/2)

Answer 1

{(1 - cosx) + sinx}/{(1 + cosx) + sinx}

1 - cosx = 2*sin²(x/2)
1 + cosx = 2*cos²(x/2)
sin(x) = 2*sin(x/2)*cos(x/2)

{2*sin²(x/2) + 2*sin(x/2)*cos(x/2)}/{2*cos²(x/2) + 2*sin(x/2)*cos(x/2)}

= 2*sin(x/2){sin(x/2) + cos(x/2)}/[2*cos(x/2){cos(x/2) + sin(x/2)}]

= sin(x/2)/cos(x/2) = tan(x/2)