To differentiate #secx# here'/ how it goes:
#secx=1/cosx#
You shall apply a quotient rule: that is #"denominator(cosx)"xx"derivative of numerator"(1)-"derivative of denominator(cosx)numerator"xx"derivative of denominator"(cosx)#
AND ALL THAT # -:("denominator")^2#
#(d(secx))/(dx)=(cosx(0)-1(-sinx))/(cosx)^2=sinx/cos^2x=1/cosx xx sinx/cosx=color(blue)(secxtanx)#
Now we go to #tanx#
Same principle as above:
#(d(tanx))/(dx)=(cosx(cosx)-sin(-cosx))/(cosx)^2=(cos^2x+sin^2x)/cos^2x=1/cos^2x=color(blue)(sec^2x)#
#color()#
Hence #color(blue)((d(secx-tanx))/(dx)=secxtanx-sec^2x)#
