I need to prove that the following equation is an identity:

Question

(1 + tan(theta/2))/(1-tan(theta/2))=tan(theta)+sec(theta)

Answer 1

For easier typing, let Ø/2 = x
then prove:
(1+tanx)/(1-tanx) = tan 2x + sec 2x

LS = (1 + sinx/cosx)/(1 - sinx/cosx)
multiply by cosx/cosx, so no change in value
= (cosx + sinx)/(cosx - sinx)

RS = sin 2x/cos 2x + 1/cos 2x
= (sin 2x + 1)/cos 2x
= (2sinxcosx + sin^2 x + cos^2 x) / (cos^2 x - sin^2 x)
= (cosx + sinx)^2 /((cosx + sinx)(cosx - sinx) )
= (cosx + sinx)/(cosx - sinx)
= LS