prove tan(x/2) = (1-cosx)/(sinx)

Question

There is the evaluation above on your website but I am unaware of the mathematical logic behind it. Could you please explain. 

 

Prove tan(x/2) = (1-cosx)/(sinx). I imagine it uses the t=tanx/2

Answer 1

You probably mean t = x/2 rather than t = tanx/2

 

The x = 2t and the equality becomes

 

tan(t) = (1 - cos(2t)) / sin(2t)

 

And as shown by tutor Ira S. the right side does equal tan(t)

Answer 2

This is equivalent of proving tan u = (1-cos 2u)/sin 2u because the angle on the tan is still 1/2 the angles on the right. So,

 

tan u = (1- [(1-(2sin u)^2]) / 2sin ucos u          (remember cos 2u = 1-(2sin u)^2  and sin 2u = 2cos u sin u)

 

tan u = (2sin u)^2 / 2sin ucos u 

tan u = (2sin u)sin u / 2sin ucos u 

tan u = sin u / cos  u 

tan u = tan u