Wie vereinfacht man #sqrt (1 + tan ^ 2x) #?

Mit:

#cos^2 x + sin^2 x = 1#

#tan x = sin x / cos x#

#sec x = 1/cos x#

wir finden:

#sqrt(1+tan^2 x) = sqrt(1+(sin^2 x)/(cos^2 x))#

#color(white)(sqrt(1+tan^2 x)) = sqrt((cos^2 x)/(cos^2 x)+(sin^2 x)/(cos^2 x))#

#color(white)(sqrt(1+tan^2 x)) = sqrt((cos^2 x+sin^2 x)/(cos^2 x))#

#color(white)(sqrt(1+tan^2 x)) = sqrt(1/(cos^2 x))#

#color(white)(sqrt(1+tan^2 x)) = sqrt(sec^2 x)#

#color(white)(sqrt(1+tan^2 x)) = abs(sec x)#