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