Question

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

Answer 1

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`