Come si verifica l'identità: # (1 + cos x + sinx) / (1 + cos x-sin x) = sec x + tan x #?

#(1+cosx+ sinx) / (1+cosx-sinx)=secx+tanx | * (1+cosx-sinx)#
#1+cosx+ sinx=(1/cosx+sinx/cosx)* (1+cosx-sinx)#
#=(1+cosx-sinx)/cosx+(1+cosx-sinx)*sinx/cosx#

#(1+cosx-sinx)/cosx=color(blue)(secx+1-tanx)#
#(1+cosx-sinx)*sinx/cosx=color(red)(tanx+sinx-sinxtanx)#

#1+cosx+ sinx=color(blue)(secx+1cancel(-tanx))+color(red)(cancel(tanx)+sinx-sinxtanx)#
#1+cosx+ sinx=secx+1+sinx-sinxtanx|-1-sinx#
#cosx=secx-sinxtanx|*cosx#
#cos^2x=1-sin^2x|+sin^2x#
#cos^2x+sin^2x=1#