Question

How do you prove that `(1 + tan(x/2)) / (1 - tan(x/2))= tan x + sec x`?

Answer 1

The easier and faster way is to use the Tangent half-angle substitution formulae:

`tan(x/2)=t`

`tanx=(2t)/(1-t^2)`

`cosx=(1-t^2)/(1+t^2)`. (remember that `secx=1/cosx`)

So:

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

and the second member becomes:

`(2t+1+t^2)/(1-t^2)=(1+t)^2/((1-t)(1+t))rArr(1+t)/(1-t)` that is the first member.