Tan x/2 = (1-cosx+sinx)/(1+cosx+sinx) prove this?

Question

thank uou

Answer 1

We know that Sin2x = 2tanx/(1+tan^2x)

Sinx = 2tan(x/2)/(1+tan^2(x/2))

Cos2x = (1-tan^2x)/(1+tan^2x)

Cosx = (1-tan^2(x/2))/(1+tan^2(x/2))

Let Tan(x/2) = t

Now Sinx = 2t/(1+t^2)

Cosx = (1-t^2)/(1+t^2)

Right handside identity = (1-Cosx+Sinx)/(1+Cosx+Sinx)

= [1-(1-t^2)/(1+t^2)+2t/(1+t^2)]/[1+(1-t^2)/(1+t^2)+2t/(1+t^2)]

= [(1+t^2-1+t^2 +2t)/(1+t^2)]/[(1+t^2+1-t^2+2t)/(1+t^2)]

= [2t^2+2t]/[2+2t]

= t(2t+2)/(2t+2)

= t

= Tan(x/2)

= Left hand side identity.