How do you prove #\frac { \tan x } { \sec x } + \frac { \cot x } { \csc x } = \sin x + \cos x#?

1 Answer
Dec 5, 2016

See the explanation

Explanation:

#color(red)(tanx)/color(blue)(secx)+color(green)(cotx)/color(magenta)(cscx)=sinx+cosx#

Rewrite #color(red)tanx# as #color(brown)(sinx)/color(DarkTurquoise)(cosx)#

Rewrite #color(blue)secx# as #1/color(DarkTurquoise)(cosx)#

Rewrite #color(green)cotx# as #1/color(red)(tanx)#

Rewrite #color(magenta)cscx# as #1/color(brown)(sinx)#

#(color(brown)(sinx)/color(DarkTurquoise)(cosx))/(1/color(DarkTurquoise)(cosx))+(1/color(red)(tanx))/(1/color(brown)(sinx))=sinx+cosx#

#(color(brown)(sinx)cancelcolor(DarkTurquoise)(cosx))/cancelcolor(DarkTurquoise)(cosx)+color(brown)(sinx)/color(red)(tanx)=sinx+cosx#

Rewrite #color(red)tanx# as #color(brown)(sinx)/color(DarkTurquoise)(cosx)#

#color(brown)sinx+color(brown)(sinx)/(color(brown)(sinx)/color(DarkTurquoise)(cosx))=sinx+cosx#

#color(brown)sinx+(cancelcolor(brown)(sinx)color(DarkTurquoise)(cosx))/cancelcolor(brown)sinx=sinx+cosx#

#color(brown)sinx+color(DarkTurquoise)cosx=sinx+cosx#