Tan Express

Tan Express

If z=tan(x/2), express [tan x]/[1-2sin x+cos x] in terms of z.?

thanks... please answer this guys...

let b=x/2; let s=sin(b), c=cos(b), z=s/c;
(0)then z^2=s^2/c^2 =(1-c^2)/c^2; or;
z^2*c^2=1-c^2, hence c^2=1/(1+z^2);
(1)then
tan(x) =tan(2b) = sin(2b)/cos(2b) =2*s*c/(c^2-s^2)=
=(2*s*c/c^2) / ((c^2-s^2)/c^2) =(2s/c) /(1-(s/c)^2) =
=2z/(1-z^2);
(2)then 1-2*sin(x) +cos(x) =1-2*sin(2b) +cos(2b) =
= [(c^2 +s^2 –4*s*c +c^2 –s^2)/c^2]*c^2 =
= [2 –4*s/c]*c^2 = 2*(1 –2*z)/(1+z^2)
(3)then
tan(x)/( 1-2*sin(x) +cos(x)) =(1)/(2) =
= (2z/(1-z^2)) / (2*(1 –2*z)/(1+z^2))
= z*(1+z^2) /((1-z^2) (1 –2z));

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