Wie beweisen Sie #cosX / (secX - tanX) = 1 + sinX #?

Beweisen #cos x / (sec x - tan x) = (1 + sin x)#

LHS # = cos x / ((1/cos x) - (sin x / cos x)# as #color(blue)(sec x = 1/cos x, tan x = sin x / cos x#

#=> cos x / ((1 - sin x) / cos x)# as #color(green)(cos x # ist das LCM des Nenners.

#=> cos^2 x / (1 - sin x)#

#=> = (1 - sin^2 x) / (1 - sin x)# as #color(blue)(cos^2x = 1 - sin^2x#

#=> ((1+ sin x) *color(red)(cancel (1 - sin x))) /color(red)(cancel (1 - sin x))#

#=> 1 + sin x#

QED