Question #26742

1 Answer
Cesareo R.
Aug 30, 2016

=(1+x tanx)/cos x=1+xtanxcosx

Explanation:

((x+y) sec(x+y)-xsecx)/y=(x+y)sec(x+y)−xsecxy=
((x+y)cosx-x cos(x+y))/(y cosx cos(x+y))=(x+y)cosx−xcos(x+y)ycosxcos(x+y)=
((x+y)cos x-x(cos x cosy-sin x sin y))/(y cosx cos(x+y))=(x+y)cosx−x(cosxcosy−sinxsiny)ycosxcos(x+y)=
(xcos x(1-cosy)+y cosx + x sinx sin y)/(y cosx cos(x+y))=xcosx(1−cosy)+ycosx+xsinxsinyycosxcos(x+y)=
=x/cos(x+y)((1-cosy)/y)+1/cos(x+y)+(x sinx)/(cos x cos(x+y)) siny/y=xcos(x+y)(1−cosyy)+1cos(x+y)+xsinxcosxcos(x+y)sinyy

but

lim_(y->0)(1-cosy)/y=0 and
lim_(y->0)(siny)/y=1

so

lim_(y->0)((x+y) sec(x+y)-xsecx)/y=(1+x tanx)/cos x

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