Question

prove the identities:
sec x + tan x = tan (x/2 + π/4)

Answer 1

RHS:
sin(x/2 + pi/4)/ cos(x/2 + pi/4) using addition formulae you get,
(sin(x/2)+cos(x/2))/(cos(x/2)-sin(x/2))
multiply and divide by sin(x/2)+cos(x/2)
you get (sin^2(x/2)+cos^2(x/2)+2sin(x/2)cos(x/2))/(sin^2(x/2)-cos^2(x/2))
which simplifies to
(1+sin x)/cos x= sec + tan x.
The last step i used the identities 2sin(x/2)cos(x/2)=sin x and
cos^2 x/2-sin^2 x/2=cos x

Answer 2

err typo the denominator in the solution shuld ahve been cos^2 x/2- sin^2 x/2 not the other way round.