Verifying this Identity.?

Question

How do I verify this Identity using double and half-angle identies?

tan(x/2 - pi/4) = tanx - secx

Answer 1

Use the identity
tan(A - B) = (tan(A)-tan(B)/(1+tan(A)tan(B))
to get
tan(x/2-pi/4) = (tan(x/2)-1)/(1+tan(x/2)) since tan(pi/4)=1
and then multiplying through by cos(x/2)/cos(x/2) gives
tan(x/2-pi/4) = (sin(x/2)-cos(x/2))/
(cos(x/2)+sin(x/2)).

Multiply the righthand side by (cos(x/2)-sin(x/2))/
(cos(x/2)-sin(x/2)) to get
(2sin(x/2)cos(x/2)-1)/
(cos^2(x/2)-sin^2(x/2))
= (sin(x)-1)/cos(x)
using the double angle identities
sin(2A) =2sin(A)cos(A) and cos(2A)=cos^2(A)-sin^2(A)
so we get the final result

tan(x/2 - pi/4) = tan(x) - sec(x).