1+sectheta-tan theta/1+sectheta+tantheta
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=1+secx-tanx/1+secx+tanx
=(1+1/cosx - sinx/cosx)/(1+1/cosx+sinx/cosx)
=(cosx+1-sinx/cosx)(cosx+1+sinx/cosx)
= 1+cosx-sinx/1+cosx+sinx
= 2cos²(x/2)-2sin(x/2)*cos(x/2) / 2cos²(x/2) + 2sin(x/2)*cos(x/2)
=2cos(x/2)(cos(x/2)-sin(x/2))/2cos(x/2)
(cos(x/2)+sin(x/2))
= cos(x/2)-sin(x/2)/cos(x/2)+sin(x/2)
= (cosx/2 - sin x/2)²/cos²x/2-sin²x/2
= 1- sinx / cosx
= secx - tanx
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