2011/097) Prove the identity: (1+cosx+sinx)/(1+cosx-sinx) = secx+tanx?

regular solution
(1+ cos x + sin x)/(1+ cos x - sin x)
= (1+ cos x + sin x)^2/ (( 1+ cos x + sin x)(1+ cosx - sin x))
= (1+ cos x + sin x)^2/ ((1+ cos x) ^2 - sin ^2 x)
= (1+ cos x + sin x)^2/ ((1+ 2cos x + cos ^2 x - sin ^2 x)
= (1+ cos x + sin x)^2/ ((cos ^2 x+ 2cos x + cos ^2 x)
= (1+ cos x + sin x)^2/ (2(cos^2 x + 2 cos x))
= (1+ cos x + sin x)^2/ (2 cos x( 1+ cos x))
= ( 1 + 2 cos x + 2 sin x + 2 sin x cos x + cos ^2 x + sin ^2 x)/ (2 cos x( 1+ cos x))
= ( 2 + 2 cos x + 2 sin x + 2 cos x sin x)/(2 cos x( 1+ cos x))
= 2(1+ cos x)(1+ sin x)/ (2 cos x (1+ cos x))
= (1+ sin x)/cos x
= sec x + tan x


alternatively ( simpler solution)

1+ cos x/ sin x = ( 2 cos ^2 x/2)/(2 cos x/2 sin x/2)
= (cos x/2)/ sin x/2

using compnondo dividendo
we get
(1+cosx+sinx)/(1+cosx-sinx) = (cos x/2 + sin x/2)/( cos x/2 - sin x/2)
= ( cos x/2 + sin x/2)^2 / (( cos x/2 + sin x/2)(cos x/2 - sin x/2)
= ( cos ^2 x/ 2 + sin ^2 x/2 + 2 cos x/2 sin x/2)/(cos^2 x/2 - sin ^2 x/2)
= (1 + sin x)/ cos x
= sec x + tan x