Prove the identity (1+cosA-sinA)/(1+cosA+sinA )= sec A - tan A

we have
(1- sin A)/cos A = (1- sin A)(1+ sin A)/(cos A (1+sin A)) = cos^2 A/(cos A(1+ sin A))
= cos A/(1+ sin A)

so
(1- sin A)/cos A = cos A/(1+ sin A)

now if a/b = c/d then a/b = (a+c)/(b+d)

so (1- sin A)/cos A = cos A/(1+ sin A) = (1- sin A + cos A)/(1+ sin A + cos A)

or (1- sin A + cos A)/(1+ sin A + cos A) = (1- sin A)/ cos A = sec A - tan A